Prepared for submission to JHEP

Decomposition of BPS Moduli Spaces and Asymptotics

of Supersymmetric Partition Functions

Arash Arabi Ardehalia and Junho Hongb

aC.N. Yang Institute for Theoretical Physics, Stony Brook University,

Stony Brook, NY 11794, USAbInstitute for Theoretical Physics, KU Leuven,

Celestijnenlaan 200D, B-3001 Leuven, Belgium

E-mail: a.a.ardehali@gmail.com, junhophysics@gmail.com

Abstract: We present a prototype for Wilsonian analysis of asymptotics of supersymmetric

partition functions of non-abelian gauge theories. Localization allows expressing such parti-

tion functions as an integral over a BPS moduli space. When the limit of interest introduces

a scale hierarchy in the problem, asymptotics of the partition function is obtained in the

Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into

various patches according to the set of light fields (lighter than the scheme dependent cut-

off Λ) they support, ii) localizing the partition function of the effective field theory on each

patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches

to obtain the final asymptotic result (which is scheme-independent and accurate as Λ→∞).

Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has

been of interest recently for its application to black hole microstate counting in AdS5/CFT4.

As a byproduct of our analysis we obtain the most general asymptotic expression for the index

of gauge theories in the Cardy-like limit, encompassing and extending all previous results.arX

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Contents

1 Introduction 1

1.1 Notation and terminology 6

2 Cardy-like asymptotics of the 4d superconformal index 6

2.1 The outer patch 8

2.2 Inner patches 14

2.3 Competition between the patches 21

2.4 The general formula 22

2.5 Example: the N = 4 index 24

2.5.1 Decomposition of the classical moduli space 24

2.5.2 The quantum moduli space and the asymptotics of the index 25

3 Effective field theory perspective 33

3.1 The 3d background 36

3.2 Light fields and cut-offs 36

3.3 The Wilsonian effective action 40

3.4 Localization in the 3d EFT and the asymptotics of the partition function 42

4 Discussion 43

4.1 Non-renormalization: Wilsonian effective action versus log I 43

4.2 Open problems 44

A Supersymmetric Chern-Simons actions 45

A.1 Chern-Simons actions from integrating out heavy fermions 46

A.2 Supersymmetrization of Chern-Simons terms 50

1 Introduction

A fruitful theme in supersymmetric quantum field theory (QFT) has been the use of exact

results to complement semi-classical reasoning. The idea is that semi-classical (or effective)

QFT should emerge as a systematic set of prescriptions to reproduce various aspects of exact

results in some—often weak-coupling and/or low-energy—limit. An outstanding landmark in

this line of research has been the use of exact supersymmetric localization results by Nekrasov

and Okounkov [1, 2] to shed light on the Seiberg-Witten solution of certain 4d N = 2 gauge

theories, originally obtained via effective field theory (EFT) arguments [3, 4].

– 1 –

In the opposite direction, semi-classical/effective QFT can be used to shed “physical”

light on the asymptotics of exact results. A major component of the present work is in this

direction, following in the footsteps of Di Pietro, Honda, and Komargodski [5, 6]. The exact

result that we consider is the 4d superconformal index [7, 8]. We study a certain Cardy-like

limit [9, 10] of the index, and obtain asymptotic expressions for it that we explain through

perturbative EFT techniques1 pioneered in [5, 6].

The superconformal index is a graded partition function of 4d N = 1 QFTs with a U(1)Rsymmetry, encoding the protected BPS spectrum of the theory, and defined explicitly as [7, 8]

I(p, q;v) = Tr (−1)F pJ1+ r2 qJ2+ r

2vqF , (1.1)

with the trace in the Hilbert space of radial quantization. Here F is the fermion number,

J1,2 are the rotation quantum numbers, and r is the U(1)R charge. The symbol vqF stands

for vq11 × · · · × vqrFrF , where va are fugacities associated to the Cartan of the flavor symmetry,

while qa are the corresponding charges. We assume |va| = 1 and 0 < |p|, |q| < 1 throughout

this paper. We will refer to va as flavor fugacities, and refer to p, q as spacetime fugacities.

Closed-form expressions are available for the index of large classes of 4d N = 1 QFTs

(see e.g. [12–15]). The closed formulas are rather complicated however, and hence often not

particularly illuminating. Substantial simplification occurs in the Cardy-like limit [9] where

the spacetime fugacities approach 1. More precisely, defining β > 0 and ω1,2 ∈ H (the upper

half-plane) via p = eiβω1 , q = eiβω2 , and defining ξa ∈ R via va = e2πiξa , we consider the limit

β → 0, with ω1,2, ξa fixed. (1.2)

This limit of the index is of physical interest for its application to testing supersymmetric

dualities [16], to studying supersymmetric gauge dynamics on R3×S1 [11], and to microstate

counting of BPS (possibly multi-center) black holes via AdS5/CFT4 [10, 17–21].

Asymptotics from the exact result

Cardy-like asymptotics of the index has been worked out in various special cases (in special

theories, or for special choices of parameters; see below for more details on relation of our

results to those of previous work). In Section 2, using direct asymptotic analysis we present the

most general expression, up to exponentially small error, for arbitrary N = 1 gauge theories

with a U(1)R symmetry and a semi-simple compact gauge group.2 We do this by employing

the exact matrix-integral expression for the index of supersymmetric gauge theories [12–14].

The matrix-integral is over a moduli space of matrix eigenvalues. It turns out that in

general on the one hand the simplest asymptotic estimate for the integrand of the index is not

1Our results regarding the interplay between exact results and EFT are hence more limited than the ones

in [1, 2] which concern non-perturbative effects. However, at least in several examples, there seems to be a

curious connection between some of the perturbative effects that we study here and non-perturbative effects

on R3×S1 [11], which might bridge our investigation to that of [1, 2]. We will not explore this connection any

further in this work.2Relaxing the semi-simplicity constraint appears to be straightforward, but we do not attempt that here.

– 2 –

uniformly valid over all of the moduli space, and on the other hand the strongest asymptotic

estimate contains too much irrelevant (exponentially suppressed) information on most of the

moduli space. This necessitates decomposing the moduli space into various patches, and using

appropriate uniform estimates on each patch. The efficient uniform grip over individual

patches then allows us to further simplify the asymptotic contribution of each patch using

some additional input from the structure of the integrand. Summing up the contributions of

all patches we obtain the asymptotics of the matrix integral as in (2.57). While the exponential

piece in (2.57) had been demonstrated earlier in special cases where the matrix-integral has

isolated saddles in the Cardy-like limit (see e.g. [16, 22]), our result that the most general

asymptotic in presence of extended saddles is that exponential dressed by a polynomial in

1/β is new.

We demonstrate the power of this method of asymptotic analysis in our case study of the

index of SU(N) N = 4 theory, where we obtain the following significant improvements over

previous results:

• For ξa = 0, the Cardy-like asymptotics of the N = 4 index was found in [16] to be of the

formf(ω1,2)βN−1 (1 + o(β0)) for the limited range of parameters ω1,2 ∈ iR>0 (i.e. p, q ∈ R).

In Section 2 we show that the ξa = 0 asymptotics is more generally for ω1,2 ∈ H given,

up to exponentially small error, by a degree N − 1 polynomial in 1/β whose leading

monomial coincides with the result of [16] when ω1,2 ∈ iR>0.

• The leading Cardy-like asymptotics of theN = 4 index in the fully-deconfined phase (see

Section 2) was obtained in [10, 17, 18] using non-uniform estimates in the integrand of

the matrix integral. But there is no guarantee in general that non-uniform estimates of

the integrand give correct asymptotics for an integral. Moreover, the papers [10, 17, 18]

assumed but did not demonstrate absence of leading-order cancellations (completely

destructive interference) due to phase oscillations in the integrand. These amount to

two significant gaps in those derivations, and the analysis in Section 2 fills both.

• In the special case with p = q, the Cardy-like asymptotics of the N = 4 index in

the fully-deconfined phase was obtained up to exponentially small error in [23].3 The

analysis in Section 2 yields the generalization of that result to p 6= q.

• The strongest results on the Cardy-like asymptotics of the N = 4 index in partially-

deconfined or confined phases were obtained in [21, 26]. In particular, in the special case

with p = q an asymptotic expression of the form (2.57) valid with exponentially small

error was obtained in [26] for SU(2) gauge group, and a similarly accurate expression

was suggested for SU(3). The analysis in Section 2 generalizes the SU(2) result of [26]

to p 6= q, and demonstrates validity of (2.57) not only for SU(3) but for arbitrary rank.

3See [24, 25] for related work studying different gauge groups.

– 3 –

Asymptotics from effective field theory

A more physical approach to finding the Cardy-like asymptotics of the index was initiated by

Di Pietro and Komargodski [5] within a Lagrangian, path-integral formulation of the index

(rather than the Hamiltonian, trace formulation as in Eq. (1.1)).

Before explaining how the Cardy-like asymptotics of the index is obtained in this ap-

proach, let us discuss how the exact matrix-integral expression for it is obtained via path-

integration in gauge theories. The 4d N = 1 gauge theory with U(1)R symmetry is placed

on a Hopf surface that is topologically S3 × S1. Turning on appropriate background fields

and deforming the Lagrangian as necessary for preserving supersymmetry [27], one uses su-

persymmetric localization [1, 28, 29] to compute the path-integral. The result is the matrix

integral expression4 for the index, up to a supersymmetric Casimir energy factor (see Sec-

tion 3 for details). The integral over the moduli space of matrix eigenvalues arises in the 4d

localization calculation as the integral over the moduli space of BPS field configurations (or

the BPS moduli space, for short).

The Cardy-like limit p, q → 1 corresponds to shrinking the circle (see Section 3). To

study the asymptotics of the index in the small-circle limit, we write the Hopf surface metric

in a Kaluza-Klein (KK) form as an S1 fiber over a three-manifold M3. The inverse of the

circle size β sets a large scale Λ ∝ 1/β in the problem, and one can use a 3d effective field

theory on M3 with only the KK fields lighter than Λ as the dynamical degrees of freedom,

and with the effect of the heavy KK fields encoded in the effective action of the light 3d

fields. The partition function of the 3d EFT should reproduce—essentially by definition of

an EFT—the large-Λ (hence small-β, Cardy-like) asymptotics of the 4d partition function.

Therefore the Cardy-like asymptotics of the index should be governed by the effective action

of the 3d EFT on M3.

The general considerations of the previous paragraph are sharpened and made computa-

tionally efficient when complemented with the following ideas.

1. The effective action contains a one-loop piece, which can be computed by first fixing

the coefficients of certain Chern-Simons (CS) terms that are easily under control, and

then supersymmetrizing those CS terms.

This was the breakthrough idea of the pioneering work of Di Pietro and Komargodski [5].

Using CS actions involving only background fields, the exponential asymptotics of the index

was found in [5] in some special cases with ξa = 0.5

Even restricting to ξa = 0, however, the asymptotic obtained in [5] is not valid for general

theories [6, 16, 31] or for general choices of the parameters ω1,2 (cf. [22]). To fix that, the

following crucial idea was provided by Di Pietro and Honda in [6].

4The localization computation is done in [29] for real-valued p, q, but the result can presumably be analyt-

ically continued—at least to a neighborhood of the real line(s).5See [10, 20, 30] for related work pertaining to ξa 6= 0.

– 4 –

2. Besides the CS terms studied in [5] involving only the background fields, one must

include in the one-loop part of the effective action also the CS terms involving the

dynamical (i.e. light) gauge fields of the 3d EFT.

With this added ingredient, the work [6] found expressions for the exponential asymptotics of

the index that are valid for arbitrary gauge theories with a U(1)R symmetry and a compact

semi-simple gauge group. More precisely, the derivation of [6] was valid only for the limited

range of parameters ω1,2 ∈ iR>0 (cf. [16] where ξa are set to zero but can be straightforwardly

restored). But the limitation ω1,2 ∈ iR>0 was purely technical, and due to the Hopf surface

background of [6] having real-valued p, q. This limitation can be overcome using the more

general background with ω1,2 ∈ H discussed in an elegant recent paper by Cassani and

Komargodski [30]. We will heavily rely on the background of [30] in Section 3.

Once dynamical fields are included in the effective action as in [6], they should be inte-

grated over. In [6], instead of performing the path-integral the dynamical fields were set to

their BPS values and then an integral over the BPS moduli space was introduced. While this

is natural and yields correctly the exponential asymptotics of the index, it begs for a sharp

justification and is in fact not sufficient for obtaining the complete subleading asymptotics

with exponentially small error. A more systematic approach was proposed in [22].

3. The partition function of the 3d EFT is obtained by supersymmetric localization of its

path-integral onM3, with the effective action consisting of the one-loop piece involving

supersymmetrized CS terms, as well as a tree-level piece coming from KK expansion of

the (UV) 4d action.

This idea was implemented in [22] for a special choice6 of ξa, but the generalization to arbitrary

ξa ∈ R is straightforward. The restriction of the one-loop supersymmetrized CS terms to the

BPS locus is justified by (and follows from) supersymmetric localization in this approach.

Also, with the inclusion of the tree-level action, the subleading asymptotics of the index is

reproduced in [22] correctly7 up to exponentially small error8 in cases of interest in that work,

although not in general as we explain shortly.

In this work we add one more idea to the mix, which in our view completes the conceptual

framework—though important puzzles remain as discussed in Section 4. We make the EFT

cut-offs explicit, and emphasize that the set of light 3d fields is different on different patches

of the BPS moduli space.

6The index studied in [22] (and also [19, 20, 30]) depending on a parameter n0, is recovered in our setting

by formally associating a U(1) flavor fugacity vχ = e2πi(−n0rχ/2) to every chiral multiplet χ and taking qχ = 1.7We emphasize that the EFT derivation in [22] relied on a certain assumption regarding the supersym-

metrized gravitational CS term. We propose in Section 3 that actually a modified version of that assumption

is needed (see Eq. (A.42)).8In particular, the hyperbolic sines (resp. the exponential factor signalling the CS level) in the integrand

of the Chern-Simons partition function found in the subleading asymptotics of the SU(N) N = 4 index in [23]

arise from localization of the tree-level (resp. one-loop) piece of the 3d effective action.

– 5 –

4. The BPS moduli space should be decomposed in some suitable scheme into various

patches, each supporting its own set of light 3d fields, and hence each having its own

3d EFT. The asymptotics of the index is obtained (up to exponentially small error) by

summing the (perturbative) EFT partition functions of all patches.

The earlier work [22] had considered only two of these patches (the outer patch and in0 in

the language of Section 2). Those were sufficient for the cases of interest in that work, but

when other patches dominate the index (as may happen in partially-deconfined phases for

example) the EFT results of [22] are not enough to reproduce the correct asymptotics.

Our EFT analysis in Section 3 parallels the general asymptotic analysis of Section 2 with

exponential accuracy. It gives a crisp physical understanding of the Cardy-like asymptotics

of the index, and in particular through the analysis of light 3d fields on different parts of the

BPS moduli space clarifies why the decomposition procedure of Section 2 is so powerful: it

is—with some stretch of imagination—what Wilson would do.

1.1 Notation and terminology

Throughout this work the symbol ' means that the ratio of the two sides is 1, up to expo-

nentially suppressed error of the form O(e−1/β). More precisely

A(β) ' B(β) ifA(β)

B(β)= 1 +O(e−c/β) as β → 0, with some fixed c > 0. (1.3)

If there is dependence on extra parameters xj , we say A(β, xj) ' B(β, xj) uniformly

over a certain domain, if there is a c > 0 that works as above (except possibly for removable

singularities inA(β,xj)B(β,xj)

) for all xj in that domain.

2 Cardy-like asymptotics of the 4d superconformal index

Consider a 4dN = 1 gauge theory with a U(1)R symmetry, a semi-simple (compact Lie) gauge

group G of finite rank rG, a finite number of chiral multiplets χ in various representations of

G, and a global flavor symmetry (compact Lie) group of rank rF .

The index can be evaluated in closed form as an elliptic hypergeometric integral [12–14]:

I(p, q;v) = (p; p)rG(q; q)rG∫hcl

Dx

∏χ

∏ρχ Γe

(rχ( τ+σ

2 ) + ρχ · x+ qχ · ξ)∏

α+Γe(α+ · x) Γe(−α+ · x)

. (2.1)

The measure is defined as

Dx :=1

|W |

rG∏j=1

dxj , (2.2)

with |W | the order of the Weyl group of G. The parameters xj will be referred to as the

holonomies, and their moduli space (parametrized by x1, . . . , xrG mod 1) is denoted hcl, which

we write explicitly as

hcl = (−1

2,1

2]rG . (2.3)

– 6 –

The rG-tuple x1, . . . , xrG is denoted x, and the rF -tuple ξ1, . . . , ξrF (with ξa related to vavia va = e2πiξa) is denoted ξ. The positive roots of the gauge group are denoted α+, and

the weights of the gauge group representation of the chiral multiplet χ are denoted ρχ. The

rF -tuple flavor charge of χ is denoted qχ, and its U(1)R charge is denoted rχ. We assume

rχ ∈ (0, 2) for all χ. The special function (·; ·) is the Pochhammer symbol

(z; q) :=

∞∏k=0

(1− zqk), (2.4)

and the function Γe(·) := Γ(e2πi(·); p, q) is the elliptic gamma function [32]:

Γ(z; p, q) :=∏j,k≥0

1− z−1pj+1qk+1

1− zpjqk. (2.5)

Finally, the parameters σ, τ are defined through

p = e2πiσ, q = e2πiτ . (2.6)

For simplicity throughout this paper we restrict our attention to the special case where

the flavor fugacities va are on the unit circle. Alternatively, we take ξa ∈ R.

Defining the four real parameters b, β ∈ R>0 and k1,2 ∈ R through (cf. [30])

σ =iβ

2π(b+ ik1), τ =

iβ

2π(b−1 + ik2), (2.7)

the Cardy-like [9] limit of our interest in this work corresponds to (cf. [10])

β → 0+, with fixed b ∈ R>0, k1,2, ξa ∈ R. (2.8)

The parameters ω1,2 defined as

ω1 :=2πσ

β= i(b+ ik1), ω2 :=

2πτ

β= i(b−1 + ik2), (2.9)

will be useful below and find a natural interpretation in the 3d EFT describing the Cardy-like

limit of the index in Section 3. Note that they are finite in the limit (2.8) and inside the

upper-half plane. Below we will assume

Re( i

ω1ω2

)6= 0 and Re

( i(ω1 + ω2)

ω1ω2

)6= 0. (2.10)

This will streamline the analysis of generic theories.9

9That is because otherwise ReV out2 or ReV out

1 , defined in Eqs. (2.18a)–(2.18b) and featuring in Definition 2,

would vanish. In non-generic situations (N = 4 theory with ξa = 0 for example) it may be that ReV out2 or

ReV out1 vanish regardless of such restrictions on ω1,2, in which case those restrictions are not needed/useful,

and our analysis applies without demanding (2.10). Note in particular that in [16], since the focus was on non-

chiral theories with ξa = 0 where ReV out2 (cf. the function Qh there) vanishes, Re

(i

ω1ω2

)6= 0 was not assumed.

In fact for concreteness (though it was not necessary) it was assumed that p, q ∈ R, so that Re(

iω1ω2

)= 0.

– 7 –

We now proceed to the asymptotic analysis of the index (2.1) in the limit (2.8). Asymp-

totics of the Pochhammer symbols are elementary (see e.g. [33]):

(p; p)(q; q) ' e−2πi σ+τ24στ · 1√

−στ· e−2πiσ+τ

24 (as β → 0). (2.11)

Here the choice of branch for the square root is through analytic continuation from σ, τ ∈ iR>0

where the positive sign is picked. (So in particular for σ = τ we have 1/√−τ2 = i/τ .)

The challenging part is the integral in (2.1). Asymptotic analysis of an integral requires

uniform estimates for its integrand.10 Below we divide hcl into an “outer patch” and its

complement—consisting of various “inner patches”. We then give separate asymptotic esti-

mates for the elliptic gammas to be used with uniform validity on each patch. Asymptotics of

the index is determined by the sum of the contributions from the individual patches. As the

reader might anticipate, the competition between the various patches can lead to interesting

phase transitions [21].

2.1 The outer patch

The simplest all-order estimate that we can use for the elliptic gammas in the integrand of

(2.1) is (cf. [16, 22, 36])

Γe((pq)

r2 e2πix

)' exp

(−2πi

(1

στ

B3(x)

3!+

1

στ

(σ + τ

2

)(r − 1)

B2(x)

2!

+3(r − 1)2(σ + τ)2 − (σ2 + τ2)

24στB1(x)

+((r − 1)3 − (r − 1)

)(σ + τ)3

48στ+ (r − 1)

σ + τ

24

)),

(2.12)

valid for any r ∈ R, and point-wise for x ∈ R \Z. The functions B1,2,3 above are the periodic

Bernoulli polynomials, explicitly given by

B3(x) := B3({x}) =1

2{x}(1− {x})(1− 2{x}),

B2(x) := B2({x}) = −{x}(1− {x}) +1

6,

B1(x) :=

{B1({x}) = {x} − 1

2 for x /∈ Z,0 for x ∈ Z,

(2.13)

with {·} := · − b·c the fractional part function (SawtoothWave[·] in Mathematica). Note that

B3, B1 are odd, while B2 is even.

We call (2.12) the “outer estimate”, because it applies uniformly for x in compact subsets

of R \ Z. That is, it applies outside an open neighborhood of Z. To be more precise,

10Alternatively, one may be able to integrate (using residue calculus in particular) and then asymptotically

analyze. See [23, 26, 34, 35] for such analyses in the present context.

– 8 –

we can pick a small positive number ε, and indicate the said open neighborhood of Z by

minn∈Z(|x−n|) < ε. The outer patch of the elliptic gamma function in (2.12) is then specified

by

minn∈Z(|x− n|) ≥ ε. (2.14)

The estimate (2.12) is uniformly valid over this outer patch. On the other hand, the open

ε-neighborhood of an integer can be called an inner patch, and the estimate (2.12) is not

uniformly valid there.

To adapt the preceding discussion to the product of gamma functions in the integrand of

the index in (2.1), we begin by defining the “singular set” S as

Sg :=⋃α+

{x ∈ hcl|α+ · x ∈Z}, Sχ :=⋃ρχ 6=0

{x ∈ hcl| ρχ · x+ qχ · ξ ∈ Z},

S :=⋃χ

Sχ ∪ Sg.(2.15)

In the next section the singular set will be interpreted as the subset of the 3d N = 2 EFT

Coulomb branch where charged massless fields arise. In the present section the significance

of S is that in its neighborhood the outer estimate (2.12) loses its uniform validity.

Denote an open ε-neighborhood of the singular set by Sε. The outer patch of the index

(2.1), which we denote by S ′ε, is defined as the complement of Sε in hcl. More explicitly we

define it as follows.

Definition 1 The outer patch S ′ε is the subset of hcl in which for all nonzero ρχ we have

minn∈Z(|ρχ · x+ qχ · ξ − n|) ≥ ε, and also for all α+ we have minn∈Z(|α+ · x− n|) ≥ ε.

This is the patch where the outer estimate (2.12) can be applied to all the gamma functions

in the index (2.1) (except possibly those with ρχ = 0 on which we comment below).

The actual numerical value of ε does not matter at this stage, although for convenience

we assume it to be small enough.11 Fixing a particular value for ε can be thought of as a

choice of “scheme”, and we will frequently see that unnecessary technical difficulties can be

avoided in schemes with small enough ε.

To compute the asymptotic contribution of the outer patch to the index (2.1)—that is

the contribution from integrating over S ′ε ⊂ hcl—we first consider generic qχ · ξ. Then the

outer estimate (2.12) can be applied to the gamma functions with ρχ = 0, if present, as well.

(We will relax this genericity assumption below.) Then using (2.11) and (2.12) we find

Iout(p, q;v) ' eβEsusy

∫S′ε

Dx

(√−ω1ω2)rG

(2π

β

)rG e−V out(x) (for generic qχ · ξ). (2.16)

11For any given 4d gauge theory with a fixed set of ξa, there exists a positive number εc such that the

qualitative structure of the outer patch S ′ε—as well as that of the inner patches discussed below—is similar for

all ε < εc. (For example, take SU(2) N = 4 theory where hcl consists of x1 ∈ (− 12, 12], and consider the simple

case where ξ1,2,3 = 0, so that S = {0, 12}. Then the outer patch is empty for ε > 1

2, while it is non-empty with

two connected components for ε < 12. See Figure 1. Therefore in this simple case εc = 1

2.) To avoid undue

complications we always take ε to be less than εc.

– 9 –

The potential V out is given by

V out(x) =V out

2 (x)

β2+V out

1 (x)

β+ V out

0 (x), (2.17)

with

V out2 (x) =

(2π)3i

ω1ω2

∑χ

∑ρχ

B3

(ρχ · x+ qχ · ξ

)3!

, (2.18a)

V out1 (x) =

(2π)2i

ω1ω2

(ω1 + ω2

2)

(∑χ

∑ρχ

(rχ − 1)B2

(ρχ · x+ qχ · ξ

)2!

+∑α

B2

(α · x

)2!

), (2.18b)

V out0 (x) =

2πi

ω1ω2

∑χ

∑ρχ

((ω1 + ω2

)28

(rχ − 1)2 − ω21 + ω2

2

24

)B1

(ρχ · x+ qχ · ξ

). (2.18c)

Finally, the supersymmetric Casimir energy in (2.16) is given by (cf. [37])

Esusy = −i(ω1 + ω2)3

48ω1ω2

(TrR3 − TrR

)− iω1 + ω2

24TrR. (2.19)

Some general remarks are in order.

First, the outer patch S ′ε is a union of finitely many, disjoint, convex polytopes Pεn.12 On

each Pεn the functions V out2,1,0 are analytic. Their non-analyticity occurs across the singular set

S which is not part of S ′ε.Next, although B3 is piecewise cubic, thanks to the (gauge)3 anomaly cancellation the

potential V out2 is piecewise quadratic—and hence quadratic on each Pεn. Similarly, because

of the U(1)R-(gauge)2 anomaly cancellation V out1 is piecewise linear. Also, since we are

considering semi-simple gauge groups, for a chiral multiplet χ in any representation we have∑χ ρ

χ = 0, and therefore V out0 is piecewise constant.

Finally, note that Iout has in general a complicated dependence on ε (i.e. “scheme de-

pendence”) through the range of integration in (2.16). However, it is not difficult to see

that when ReV out2 is minimized strictly within the outer patch, the sensitivity of Iout to ε is

exponentially small. This is because the integral in (2.16) is determined, up to exponentially

small error, by the contribution from a small neighborhood of the locus of minima of ReV out2 .13

Let us now consider a simple such scenario, where ReV out2 is minimized on a single point

x∗ ∈ Pεn∗ ⊂ S′ε, and evaluate Iout. We can of course restrict the integration domain in

(2.16) to Pεn∗ (or in fact any compact subset of it containing x∗) because our assumption

that ReV out2 is minimized at x∗ implies that the integrand is exponentially suppressed away

from x∗. Moreover, on Pεn∗ ⊂ S′ε the Taylor expansions of V out

2,1,0 are exact. Therefore using

12The polytopes Pn in [16] are related to our Pεn as limε→0 Pεn = Pn.13As we will discuss momentarily, when this locus is extended, the dominant contribution to the asymptotics

of the index comes from near the subset of it where ReV out1 is minimized.

– 10 –

∂xjVout

2 (x∗) = 0 and B′j(x) = jBj−1(x) we can estimate the integral (2.16) as

Iout(p, q;v) ' eβEsusy e−Vout∗ Z3d

n∗(ω1, ω2;2πε

β), (2.20)

where V out∗ := V out(x∗), and

Z3dn∗(ω1, ω2;

2πε

β) :=

∫2πβPεn∗

Dx

(√−ω1ω2)rG

e− 2πiω1ω2

koutij

xixj2− 2πiω1ω2

(ω1+ω2

2)koutjR xj . (2.21)

Here xj := 2πβ (xj−x∗j ), and we have denoted the rescaled polytope by 2π

β Pεn∗ . The coefficients

koutij and kout

jR are given by

koutij =

∑χ

∑ρχ

B1

(ρχ · x∗ + qχ · ξ

)ρχi ρ

χj ,

koutjR =

∑χ

∑ρχ

B1

(ρχ · x∗ + qχ · ξ

)ρχj (rχ − 1).

(2.22)

Note that our assumption that ReV out2 is minimized at x∗ implies that the second derivative

of ReV out2 is positive definite at x∗, and hence

Re( i

ω1ω2

)koutij (2.23)

is a positive definite matrix. Therefore by replacing the integration domain in (2.21) with

xj ∈ (−∞,∞) we would only introduce an exponentially small error. This removes the ε-

dependence of Z3dn∗ , and demonstrates the exponentially small sensitivity of Iout to ε in the

present scenario as mentioned above. The end result is the simplification of (2.20) to

Iout(p, q;v) ' eβEsusy e−Vout∗ Z3d

n∗(ω1, ω2), (2.24)

where

Z3dn∗(ω1, ω2) := Z3d

n∗(ω1, ω2;∞) =

∫ ∞−∞

Dx

(√−ω1ω2)rG

e− 2πiω1ω2

koutij

xixj2− 2πiω1ω2

(ω1+ω2

2)koutjR xj . (2.25)

We emphasize that (2.24) is valid for isolated x∗ in the outer patch (and for qχ · ξ /∈ Z, or

more generally in absence of light zero weights for chiral multiplets with qχ · ξ ∈ Z).

If ReV out2 is minimized on a finite number of isolated points within the outer patch, each

would contribute to Iout as in (2.24), therefore to get the total contribution we should replace

Z3dn∗(ω1, ω2) on the RHS of (2.24) with

∑n∗Z3dn∗(ω1, ω2).

If ReV out2 is minimized not at isolated points but on an extended locus within the outer

patch, then Iout is determined, up to exponentially small error, by the contribution from near

the subset of that extended locus where ReV out1 is minimized. We denote this subset by hqu.

If the latter consists of an isolated point, then the derivation of (2.24) still applies (and if

there is a degeneracy, Z3dn∗ on the RHS of (2.24) should be replaced with

∑n∗Z3dn∗ as in the

– 11 –

previous paragraph). Now koutij becomes positive semi-definite, with those entries associated

with the flat directions of ReV out2 being zero.Still, Z3d

n∗(ω1, ω2) remains well-defined, as koutjR

guarantee exponential decay of the integrand along the flat direction of ReV out2 in the present

scenario.

If the locus of minima of ReV out2 is extended within the outer patch, and the subset of it

where ReV out1 is minimized—denoted hqu—is also extended, then the derivation of (2.24) fails!

This is because the integrand of (2.25) does not decay along the flat directions of both ReV out2

and ReV out1 , so the integration domain in (2.25) cannot be replaced with xj ∈ (−∞,∞). In

other words, now the asymptotics of Iout is sensitive to ε. As we will see, this sensitivity

turns out to be in an overall O(β0) multiplicative factor in (2.20), so let us for the moment

consider only the singular asymptotics of log Iout which remains insensitive to ε. We denote

the number of flat directions by dimhqu. (We will give a more precise definition of dimhqu

below.) Then Z3dn∗(ω1, ω2; 2πε

β ) would have a

( 1

β

)dimhqu (2.26)

divergence from integrating along the flat directions. Denoting the value of V out1,2 on hqu by

V out1,2∗, from (2.17) and (2.20) we obtain

Iout(p, q;v) ≈ e−V out2∗β2 e

−Vout1∗β( 1

β

)dimhqu , (2.27)

up to an O(β0) error upon taking logarithms of the two sides.

To have the result (2.27) apply equally well to all the scenarios in the previous four

paragraphs, we define hqu (the “quantum moduli space”) and dimhqu in general as follows.

Definition 2 Fix ω1,2, ξa, and consider the locus of minima of ReV out2 (as in (2.18a)) in-

side hcl. The set hqu is defined as the subset of that locus where ReV out1 (as in (2.18b)) is

minimized. If hqu is a finite set, dimhqu is defined to be zero. If hqu is extended, it would

consist of multiple, possibly intersecting, flat elements inside hcl, and dimhqu is defined to be

the dimension of the flat element(s) with largest dimension.

The claim in the above definition regarding the structure of hqu when it is extended can

be demonstrated as follows. Recall that on each component of hcl \ S the potentials V out1,2

are analytic. Let us denote connected components of hcl \ S by Pn := limε→0 Pεn. So V out2

is quadratic on each Pn, and V out1 is linear. The locus of minima of ReV out

2 inside each

component is hence determined by linear relations arising from differentiating the quadratic

expression for ReV out2 on Pn, and the subset of this locus where ReV out

1 is minimized is

similarly a linear subset. Therefore hqu consists of flat (or linear) subsets of hcl \ S, as well as

their extension to S (because of the continuity of V out1,2 ). There may be other subsets of S that

are contained in hqu too (besides the subsets arising from extension of hqu ∩ (hcl \ S) to S),

and those will similarly have a flat (linear) structure, as S is also a union of flat hyperplanes.

– 12 –

We therefore conclude that hqu consists of finitely-many, possibly intersecting, flat elements

inside hcl, as claimed.14

Equipped with this refined knowledge of the structure of hqu, we can now proceed to

improve (2.27) to exponential accuracy. Let us denote by Pεn∗ those outer-patch polytopes

that intersect hqu. Similarly to how we derived (2.20) we get

Iout(p, q;v) ' eβEsusy e−Vout∗∑n∗

Z3dn∗(ω1, ω2;

2πε

β). (2.28)

Here Z3dn∗(ω1, ω2; 2πε

β ) is precisely as before (see (2.21)), and V out∗ stands for V out evaluated on

hqu. Now we first note that for each n∗ the integral in (2.21) in the directions perpendicular to

hqu gives a multiplicative β-independent contribution.15 To write down the contribution from

the directions along hqu, let us denote the dimension of Pεn∗ ∩ hqu by dimhn∗qu. The directions

along hqu contribute to Z3dn∗(ω1, ω2; 2πε

β ) in (2.21) as

∣∣2πβ

(Pεn∗ ∩ hqu

)∣∣(√−ω1ω2)dimhn∗qu

, (2.29)

because koutij and kout

jR are zero along hqu (thanks respectively to V out2 and V out

1 being flat

along it). The symbol∣∣2πβ (Pεn∗ ∩ hqu)

∣∣ stands for the volume of the re-scaled set; it can be

alternatively written as(

2πβ

)dimhn∗qu∣∣Pεn∗ ∩ hqu

∣∣. We can hence simplify (2.28) to

Iout(p, q;v) ' eβEsusy e−Vout∗( dimhqu∑

j=0

Coutj (ε)

βj). (2.30)

The coefficients Coutj (ε) above depend on ε, as well as on ω1,2 and ξa. The term

Coutj (ε)

βjin the

sum arises16 from those n∗ for which dimhn∗qu = j. If there is no such n∗, then Coutj (ε) is zero

of course. The highest term in the sum corresponding to j = dimhqu recovers (2.27).

We emphasize that (2.30) is valid only if hqu intersects the outer patch. Otherwise, as we

will see below, the contribution of the outer patch is exponentially suppressed compared to the

patches (discussed below) that do intersect hqu, so Iout would be negligible with exponentially

small error.

14What we call a flat (or linear) subset of hcl, can be alternatively thought of as a convex polytope inside

hcl, thanks to the half-space representation (or H-description) of convex polytopes. See the Wikipedia article

on Convex Polytope.15This is implied by an argument similar to the one leading to (2.24): the only β-dependence of

Z3dn∗(ω1, ω2; 2πε

β) is through the cut-off in (2.21), which can be removed (with exponentially small error) in

the directions perpendicular to hqu where the integrand in (2.21) is exponentially suppressed.16Here we are assuming that all these allowed coefficients are nonzero. That is, they do not vanish due to

“unnatural” cancellations. This should of course be checked in specific examples.

– 13 –

2.2 Inner patches

A stronger all-order estimate for the elliptic gamma function is (cf. [16, 36])

Γe((pq)

r2 e2πix

)' exp

(−2πi

(1

στ

K3(x)

3!+

1

στ

(σ + τ

2

)(r − 1)

K2(x)

2!

+3(r − 1)2(σ + τ)2 − (σ2 + τ2)

24στK1(x)

+((r − 1)3 − (r − 1)

)(σ + τ)3

48στ+ (r − 1)

σ + τ

24

))× Γh

(2π

βxZ +

(ω1 + ω2

2

)r ;ω1, ω2

),

(2.31)

valid for any r ∈ R, and point-wise for any x ∈ R. Here Γh(·;ω1, ω2) is the hyperbolic gamma

function [36], while Kj(x), which we call modified periodic Bernoulli polynomials, are defined

as

Kj(x) := Bj(x) +j

2sign(xZ)(xZ)j−1. (2.32)

Here xZ := x−nint(x), with nint(·) the nearest integer function (Round[·] in Mathematica).17

The functions Kj have three notable properties. First, unlike Bj , the functions Kj are

smooth across Z. In other words, the second term on the RHS of (2.32) kills the part of Bj

that is non-analytic across Z. This can be seen explicitly in the domain −12 < x < 1

2 where

B3(x) = x3 − 3

2x|x|+ x

2, K3(x) = x3 +

x

2, (2.33)

B2(x) = x2 − |x|+ 1

6, K2(x) = x2 +

1

6, (2.34)

B1(x) = x− sign(x)

2, K1(x) = x. (2.35)

The non-analyticity of Kj occurs instead across Z + 12 . Second, since for integer x we have

xZ = 0, the two functions Kj and Bj coincide on Z. Third, as can be confirmed from the

above explicit expressions in the domain −12 < x < 1

2 , we have

( d

dx

)j−1Kj(x)

∣∣Z = 0. (2.36)

We call (2.31) the “inner estimate”, because it applies uniformly for x inside the open

neighborhoods of Z where the outer estimate is not uniformly valid. In fact, the inner estimate

(2.31) applies uniformly over all x ∈ R. But for x outside an ε-neighborhood of Z the argument

17In principle, we can resolve the ambiguity in Kj(x) at x ∈ Z+ 1/2 either like Mathematica by picking the

even number for nint, or by defining Kj(x) to be the average of its left and right limits there. This will not

be necessary for our purposes though, as outside a small (ε-) neighborhood of Z we stop using (2.31) and use

the outer estimate (2.12) instead.

– 14 –

of the hyperbolic gamma function diverges, and one can use the asymptotic formula (see [36])

Γh(2π

βx+

(ω1 + ω2

2

)r ;ω1, ω2

)' exp

(2πi

(1

στ

32x|x|

3!+

1

στ

(σ + τ

2

)(r − 1)

|x|2!

+3(r − 1)2(σ + τ)2 − (σ2 + τ2)

24στ

sign(x)

2

)),

(2.37)

which together with (2.32) simplifies the inner estimate (2.31) back to the outer one (2.12).

In other words, the inner estimate is the master estimate: it reproduces the outer estimate

(2.12) for x in compact subsets of R\Z, but for x inside open neighborhoods of Z of the form

minn∈Z(|x− n|) < ε, (2.38)

and in particular inside O(β) neighborhoods of Z (where the argument of the hyperbolic

gamma function in (2.31) is finite), it contains more information than the outer estimate.

Now, when do we use the inner estimate (2.31) for the gamma functions in the integrand

of the index (2.1)? The following definition addresses this question.

Definition 3 The set Sε is the subset of hcl in which either for some nonzero ρχ we have

minn∈Z(|ρχ ·x+qχ ·ξ−n|) < ε, or for some α+ we have minn∈Z(|α+ ·x−n|) < ε. We refer to

such weights and roots as light. Let us include among the light weights also all the zero weights

of any chiral multiplet χ for which minn∈Z(|qχ · ξ − n|) < ε.18 Denoting the corresponding

minimizing integer of a light weight ρχ (or a light root α+) by nρχ (or nα+), we refer to the

pair (ρχ, nρχ) (or (α+, nα+)) as a light mode. We decompose Sε into finitely-many, non-

intersecting patches in0, in1, . . . , to be referred to as the inner patches, distinguished by their

differing set of light modes. The inner patch in0 is the one containing the origin x = 0.

Note that all roots are light within in0. In general, all weights of chiral multiplets χ

satisfying qχ · ξ ∈ Z are also light inside in0. An interesting special case is when all qχ · ξ are

generic qχ · ξ 6∈ Z; then in0 would have no light weights.

Consider the nth inner patch inn, and denote its set of light roots and weights by Ln.

The other roots and weights of the index (2.1) comprise a set that we denote by Hn. We

apply the inner estimate (2.31) to the weights and roots in Ln, and use the outer estimate

(2.12) for those in Hn. This way we obtain the asymptotic contribution of the patch inn to

the index given in (2.1) as

Iinn(p, q;v) ' eβEsusy

∫inn

Dx

(√−ω1ω2)rG

(2π

β

)rG e−V inn (x)

×∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Ln Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) ,

(2.39)

18Note that by taking ε small enough we can ensure that only chiral multiplets with qχ · ξ ∈ Z have such

light zero weights. To avoid undue complications, we always take ε as small as needed to simplify the analysis.

We hence assume from now on that only the chiral multiplets with qχ · ξ ∈ Z have light zero weights.

– 15 –

with Esusy as before (see (2.19)). The potential V inn is given by

V inn(x) =V inn

2 (x)

β2+V inn

1 (x)

β+ V inn

0 (x), (2.40)

with

V inn2 (x) =

(2π)3i

ω1ω2

∑χ

[ ∑ρχ∈Hn

B3

(ρχ · x+ qχ · ξ

)3!

+∑ρχ∈Ln

K3

(ρχ · x+ qχ · ξ

)3!

], (2.41a)

V inn1 (x) =

(2π)2i

ω1ω2

(ω1 + ω2

2

)(∑χ

(rχ − 1)

[ ∑ρχ∈Hn

B2

(ρχ · x+ qχ · ξ

)2!

+∑ρχ∈Ln

K2

(ρχ · x+ qχ · ξ

)2!

]

+∑α∈Hn

B2

(α · x

)2!

+∑α∈Ln

K2

(α · x

)2!

), (2.41b)

V inn0 (x) =

2πi

ω1ω2

∑χ

((ω1 + ω2

)28

(rχ − 1)2 − ω21 + ω2

2

24

)×[ ∑ρχ∈Hn

B1

(ρχ · x+ qχ · ξ

)+∑ρχ∈Ln

K1

(ρχ · x+ qχ · ξ

)]. (2.41c)

We have boxed Eq. (2.39) because everything else regarding the Cardy-like asymptotics

of the index follows from it, either as a special case or through elementary manipulations

of asymptotic analysis. In particular, the expression (2.16) for Iout, valid for generic qχ · ξ,

follows from it as the “special case” with Ln = ∅. More generally, for qχ · ξ not necessarily

generic, we can obtain the correct expression for Iout from (2.39) by letting Ln consist of the

zero weights (ρχ = 0) of chiral multiplets χ for which qχ · ξ ∈ Z (as well as replacing the

integration domain inn with the outer patch S ′ε of course).

Some general remarks are now in order.

First, note that V inn2,1,0 in (2.41) are obtained from V out

2,1,0 in (2.18) by replacing the Bj

functions associated to the light weights and roots with Kj functions. An important conse-

quence of this replacements is that even though the inner patch inn intersects the singular

set S, the functions V inn2,1,0 are in fact smooth on inn. This is because inn, by construction,

intersects S where the arguments of K3,2,1 in V inn2,1,0 become integers while those of B3,2,1 are

away from integers; as mentioned below (2.32) the functions Kj(x) are analytic across x ∈ Z,

while B3,2,1 are analytic away from integers.

Next, according to their definition in (2.32), the functions Kj(x) do not differ from Bj(x)

in their highest power, that is xj . Consequently, the highest powers in V inn2,1,0 cancel similarly to

what we had for V out2,1,0. To be specific, V inn

2 and V inn1 are respectively quadratic and linear on

inn thanks to the (gauge)3 and U(1)R-(gauge)2 anomaly cancellations, while V inn0 is constant

because of the gauge group being semi-simple.

– 16 –

Finally, in general Iinn has a complicated dependence on ε through the range of integration

in (2.39). However, when ReV out2 is minimized on a single point x∗ ∈ inn, the sensitivity to

ε turns out to be exponentially small and we can evaluate Iinn as follows.

We can assume x∗ is on the singular set S, and furthermore that the arguments of K3,2,1

in V inn2,1,0 are all integers at x∗:

ρχ · x∗ + qχ · ξ , α · x∗ ∈ Z for all ρχ, α ∈ Ln. (2.42)

This is because otherwise by taking ε small enough we can shrink inn so that x∗ falls out of

it. (Recall that while ε is a finite number, we frequently take it as small as needed to simplify

the analysis.)

Noting that Kj coincide with Bj on integers, and that the substitution Kj → Bj leads

to V inn → V out, we deduce that

V inn(x∗) = V out(x∗). (2.43)

Taylor expanding the potentials V inn2,1,0 around x∗, and using ∂xjV

out2 (x∗) = 0 as well as (2.36),

we can estimate the integral (2.39) as

Iinn(p, q;v) ' eβEsusy e−Vout(x∗) Z inn

3d (ω1, ω2;2πε

β), (2.44)

with

Z inn3d (ω1, ω2;

2πε

β) :=

∫2πβ

inn

Dx

(√−ω1ω2)rG

e− 2πiω1ω2

kinnijxixj2− 2πiω1ω2

(ω1+ω2

2) kinnjR xj

×∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + ρχ · x)∏

α+∈Ln Γh(α+ · x

)Γh(− α+ · x

) . (2.45)

Here xj := 2πβ (xj − x∗j ), and we have denoted the rescaled patch by 2π

β inn. The coefficients

kinnij and kinn

jR are given by

kinnij =

∑χ

∑ρχ∈Hn

B1

(ρχ · x∗ + qχ · ξ

)ρχi ρ

χj ,

kinnjR =

∑χ

∑ρχ∈Hn

B1

(ρχ · x∗ + qχ · ξ

)ρχj (rχ − 1).

(2.46)

For large xj , we can use the asymptotics of the hyperbolic gamma function in (2.37) to see

that the integrand of (2.45) simplifies back to that of (2.21). (There might be additional

contributions from the hyperbolic gamma functions with light zero weights ρχ = 0 which do

not simplify at large xj , but those are independent of xj anyway and can be taken outside

the integral in (2.45).) So our assumption that ReV out2 is minimized at x∗ implies again that

by replacing the integration domain in (2.45) with xj ∈ (−∞,∞) we would only introduce

exponentially small error. This removes the ε-dependence of Z inn3d , and demonstrates the

– 17 –

exponentially small sensitivity of Iinn to ε in the present scenario as mentioned above. The

end result is the simplification of (2.44) to

Iinn(p, q;v) ' eβEsusy e−Vout(x∗) Z inn

3d (ω1, ω2), (2.47)

where

Z inn3d (ω1, ω2) := Z inn

3d (ω1, ω2;∞) =

∫ ∞−∞

Dx

(√−ω1ω2)rG

e− 2πiω1ω2

kinnijxixj2− 2πiω1ω2

(ω1+ω2

2) kinnjR xj

×∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + ρχ · x)∏

α+∈Ln Γh(α+ · x

)Γh(− α+ · x

) .(2.48)

If there are flat directions, the derivation of (2.47) fails, because the integrand of (2.45)

would not decay along the flat direction. Similarly to how we derived (2.30) however, we now

argue that a similar expression applies here.

Without singular intersections

First assume that inn contains parts of a number of hyperplanes S1, . . . ,Sm, . . . belonging to

the singular set S, but does not contain any intersections of them. That is, inn ∩ S consists

of inn ∩S1, . . . , inn ∩Sm, . . . that do not intersect, and are hence at O(ε) distances from each

other.

Write the asymptotic contribution of inn to the index as∫

innIinn , where the integrand

can be read from (2.39), and write the asymptotic contribution of the outer patch to the index

as∫

out Iout, where the integrand can be read for generic qχ · ξ from (2.16) or more generally

from (2.39) as explained below (2.41). Now write∫inn

Iinn =

∫inn

Iout +

∫inn

(Iinn − Iout). (2.49)

It should be clear from our earlier discussion of the outer patch contribution that the first

integral on the RHS localizes with exponential accuracy onto inn ∩ hqu. It then contributes

to the index an expression similar to (2.30).

To estimate the second integral on the RHS of (2.49), first note that Iinn approaches Iout

exponentially fast as we move away from S. (Because the inner estimate recovers the outer

estimate exponentially accurately away from Z.) Therefore the second integral localizes with

exponential accuracy to inn ∩S, which we can write alternatively as inn ∩ (S1 ∪ · · · ∪ Sm . . . ).Next, consider the contribution from a small neighborhood of one of the components

of inn ∩ S, say the component inn ∩ Sm. We denote the small neighborhood by inn ∩ Sε1m ,

with ε1 chosen small enough such that inn ∩ Sε1m does not intersect any other component

of inn ∩ S. In this region Iinn is approximated exponentially accurately by what we denote

by Im. The latter is obtained from (2.39) by replacing the set Ln with the set of the light

weights/roots associated to Sm, which we denote by Lm. This replacement entails in particular

– 18 –

a replacement V inn → V m, where V m is obtained from (2.41) by the replacement Ln → Lm,

as well as Hn → Hm, where Hm consists of all the other weight/roots of the index that are not

in Lm. The second integral on the RHS of (2.49) can then be expressed up to exponentially

small error as∫inn

(Iinn − Iout) '∑m

∫inn∩S

ε1m

(Im − Iout) =∑m

∫inn∩S

ε1m

Iout

( ImIout− 1)

=∑m

eβEsusy

∫inn∩S

ε1m

Dx

(√−ω1ω2)rG

(2π

β

)rG e−V out(x) gm(2πx⊥m

β

),

(2.50)

where

gm(2πx⊥m

β

):=

ImIout−1 = e−V

m(x)+V out(x)

∏χ

∏ρχ∈Lm Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Lm Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) −1.

(2.51)

Above we have used the coordinate x⊥m which measures the distance from Sm.19 Note that

gm(2πx⊥m

β

)decays to zero exponentially fast for any finite x⊥m, because Im approaches Iout

exponentially fast as we move away from Sm.

Let us assume inn ∩Sm intersects hqu; otherwise as the argument below (2.55) shows the

contribution from inn ∩Sε1m to the index is exponentially smaller than those from the patches

that do intersect hqu, and hence negligible. This assumption requires in particular that innintersects hqu. Let us denote the inner patches intersecting hqu by inn∗ .

The integrand of the mth term in (2.50) is the product of two functions: e−Vout

and

gm. Intuitively, we expect the integral to localize around inn ∩ Sm ∩ hqu, because e−Vout

is

exponentially suppressed away from hqu, while gm is exponentially suppressed away from Sm.

To be able to control the details however, we proceed as follows. We first integrate over the

constant-x⊥m slices parallel to Sm. At this stage gm can be treated as a constant, and the

analysis is quite similar to the one we performed for the outer patch around (2.28). The inte-

gral localizes onto inn∩hqu, with an integrand that is proportional to eβEsusye−Vout∗ gm

(2πx⊥mβ

).

The remaining integral over x⊥m clearly localizes onto x⊥m = 0, that is onto inn ∩ Sm ∩ hqu.

Thus the mth term in (2.50) would have asymptotics similar to (2.30), namely

mth term in (2.50) '∑jm

eβEsusye−Vout∗

Cjm(ε)

βjm, (2.52)

where Cjm(ε) is nonzero only if inn ∩ Sm ∩ hqu has a component of dimension jm. Summing

the asymptotics (2.52) over m as required in (2.50), we conclude that Iinn =∫

innIinn has the

19Implicit in writing (2.51) is the claim that the RHS is only a function of x⊥m/β. While this is obvious for

the hyperbolic gammas because of the form of their arguments, for the combination −V m(x)+V out(x) it is not

so obvious. To see why that is the case, let us consider (−V m2 +V out2 )/β2. (Similar considerations apply to the

other pieces.) Using (2.18a) and (2.41a) we have −V m2 +V out2 ∝

∑ρχ∈Lm [−K3

(ρχ ·x+qχ ·ξ

)+B3

(ρχ ·x+qχ ·ξ

)];

noting from (2.33) that the cubic and linear parts of B3 and K3 are equal, we deduce that only the quadratic

part of B3 survives in the expression for −V m2 + V out2 . Since for ρχ ∈ Lm the combination ρχ · x + qχ · ξ is

proportional to x⊥m, we conclude that (−V m2 + V out2 )/β2 is a homogeneous function of x⊥m/β of degree two.

– 19 –

same asymptotic form as in (2.30), where the polynomial factor here would only contain those

powers of 1β that correspond to the dimensions of the various components of inn ∩ S ∩ hqu.

To summarize, the first integral on the RHS of (2.49) localizes to inn ∩ hqu, while the

second integral localizes to inn ∩S ∩hqu. Both yield asymptotics of the form (2.30), implying

a similar asymptotic for the left-hand side, as claimed.

With singular intersections

Now assume that inn contains intersections of several hyperplanes in S. (We can ensure

no inner patch contains two or more singular intersection sets away from each other by,

again, taking ε to be small enough.) More explicitly, say inn intersects a number of singular

hyperplanes S1, . . . ,Sm, . . . as well as (part or all of) some intersections of those hyperplanes.

Then the following refined version of (2.49) becomes useful:

∫inn

Iinn =

∫inn

Iout +

∫inn

∑m

(Im − Iout) +

∫inn

((Iinn − Iout)−

∑m

(Im − Iout)). (2.53)

An argument similar to the one we applied to (2.50) shows that the first integral on the RHS

of (2.53) localizes to inn ∩ hqu, while the second integral localizes to ∪m(inn ∩Sm ∩ hqu), and

the last integral localizes20 to the various singular intersections inside inn ∩ hqu. Moreover,

for the two latter cases the integrand of the localized integrals coincides with Iout up to β-

independent factors (see the paragraph of (2.52)). Therefore all the three integrals on the

right-hand side of (2.53) give asymptotics of the form (2.30), implying a similar asymptotic

for the left-hand side, as claimed:

Iinn∗ (p, q;v) ' eβEsusy e−Vout∗( dimhqu∑

j=0

Cinn∗j (ε)

βj). (2.54)

We emphasize again that this asymptotic relation applies only to those inner patches inn∗that intersect hqu. An inner patch inn that does not intersect hqu has asymptotics that is

exponentially smaller than (2.54) as we argue below

An important aspect of the result (2.54) is that the singular asymptotics of log Iinn∗is

determined entirely by the outer potentials V out2,1 .

20To shows that the third integral on the RHS of (2.53) receives non-negligible contributions only from near

intersections of singular hyperplanes, let us demonstrate exponential suppression of its integrand away from

all such singular intersections: if we are also away from all singular hyperplanes, then Iinn − Iout and Ij − Iout(for all j) are separately exponentially suppressed and we are done; if we are near a hyperplane Sm, then

Iinn − Iout is not exponentially suppressed, and neither is Im− Iout, but (Iinn − Iout)− (Im− Iout) = Iinn − Imis exponentially suppressed (because Im approximates Iinn exponentially accurately if we are away from the

other singular hyperplanes) and we are done.

– 20 –

2.3 Competition between the patches

Asymptotics of the index (2.1) is obtained by combining the contributions of the outer patch

and the inner patches given in (2.16) and (2.39) respectively:

I(p, q;v) = Iout(p, q;v) +∑n

Iinn(p, q;v). (2.55)

One or several of the patches could give the dominant contribution to the index through

(2.55) in the Cardy-like limit. We now argue that these are the patches that intersect hqu.

We do this in two steps. In the first step we argue that if the outer patch does not intersect

hqu, its contribution is exponentially suppressed compared to those inner patches that do

intersect hqu. This is quite straightforward in fact, starting from (2.16) (or its generalization

for qχ ·ξ not necessarily generic, that follows from (2.39)). We simply note that the minimum

of ReV out2 over the outer patch is greater than ReV out

2∗ by a strictly positive number of order ε

or ε2. That is because V out2 (x) is piecewise quadratic in xj , and the outer patch is a distance

ε away from the singular set which we are assuming contains hqu. Therefore the integrand of

Iout is uniformly exponentially (e−1/β2type) suppressed compared to (2.54), and we conclude

that Iout is exponentially suppressed relative to Iinn .21

In the second step we argue that if an inner patch inn does not intersect hqu, its con-

tribution is exponentially smaller than the patches that intersect hqu. To demonstrate this

it would be sufficient to argue that over inn we have Iinn uniformly bounded above by a

constant multiple of Iout, because then the argument of the previous paragraph applies again

and we are done. (Note that we previously discussed uniformly valid exponentially accurate

estimates, while here we are appealing to—much weaker—bounds that are uniformly valid.)

To establish the uniform boundedness, consider the ratio

Iinn

Iout= e−V

inn (x)+V out(x)

∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Ln Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) =: ginn

(2πx⊥innβ

)+1.

(2.56)

Here x⊥inn stands for a set of new coordinates measuring distance from the singular hyperplanes

intersecting inn, and we have defined ginn analogously to (2.51). Now define S(β) as the O(β)

neighborhood of S obtained by replacing ε → Λβ2π in Sε, for some Λ > 0. Fixing a particular

positive number for Λ amounts to a choice of scheme. Now decompose inn as the union

of inn ∩ S(β) and inn ∩ (Sε \ S(β)). On the latter set, the asymptotics of the hyperbolic

gamma function (2.37) implies that by taking Λ large enough we can ensure that ginn is

arbitrarily close to zero, and therefore we get our desired uniform bound on Iinn/Iout. It

remains to establish a uniform bound on inn∩S(β). For that, we define the re-scaled variables

21More precisely, in this paragraph we have assumed that the locus of minima of ReV out2 does not interect

the outer patch. It may be that ReV out2 is minimized on a locus intersecting the outer patch, but the subset

of that locus where ReV out1 is minimized does not intersect the outer patch. (If it did, hqu would intersect the

outer patch, while we are assuming it does not.) A similar argument applies in that case too, only this time

with an e−1/β type, rather than e−1/β2

type, exponential suppression.

– 21 –

x⊥inn := 2πβ x⊥inn

. Then since the function ginn(x⊥inn) is continuous, it is uniformly bounded on

the compact domain |x⊥j | ≤ Λ, and we are done. The said continuity relies on the fact that the

poles of the chiral-multiplet gamma functions and the zeros of the vector-multiplet gamma

functions in (2.56) are avoided for |x⊥j | ≤ Λ, which can be verified assuming rχ > 0.22

We have thus established the claim that the patches dominating the index are those that

intersect the quantum moduli space hqu.

We see from (2.18a) and (2.18b) that hqu can depend on the continuous parameters

ξa, as well as the discrete parameters sign(Re( iω1ω2

)) and sign(Re( i(ω1+ω2)ω1ω2

)). These control

parameters provide the arena for the competition between the various patches in (2.55).

By changing the control parameters the dominant patches might change, and this can be

thought of as a phase transition in the index. Since β is analogous to inverse-temperature in

thermal quantum physics, the transitions that we are discussing in the β → 0 asymptotics

are analogous to infinite-temperature phase transitions.

Note that the structure of the various patches also changes by changing ξa, because Sχdepend on ξa.

2.4 The general formula

We can now restrict the sum in (2.55) to the patches intersecting hqu to obtain the asymptotics

of the index. From (2.30) and (2.54) we obtain

I(p, q;v) ' eβEsusy e−V

out2∗β2−V

out1∗β−V out

0∗( dimhqu∑

j=0

Cjβj). (2.57)

Here Cj is defined as Coutj (ε)+

∑n∗C

inn∗j (ε) if the outer patch intersects hqu, and as

∑n∗C

inn∗j (ε)

otherwise. Note that while Coutj (ε) and C

inn∗j (ε) can be ε-dependent, the coefficients Cj should

be independent of the decomposition scheme. In our way of decomposing hcl a choice of scheme

corresponds to picking a specific value for ε. Therefore Cj are independent of ε, and functions

of ω1,2 and ξa only.

While most previous papers have focused on cases with dimhqu = 0 where the polynomial

factor in (2.57) is simply a constant, non-trivial polynomial factors have been observed in the

indices of SO(3) SQCD with two flavors [16] and SU(2) N = 4 theory [26].23 The non-trivial

polynomials were discovered in [16, 26] using alternative representations of the index, which

are available in those special theories but not in general: the finding in [16] relied on the

free-fermion representation of [39] (available for a limit of the index in a particular class of

theories at present [40]), and the one in [26] used the Bethe-Ansatz approach [41, 42] which

is not fully reliable at present for rank greater than one [21]. Our demonstration of (2.57)

22To summarize, the logic of the second step is that |∫inn

Iinn | < C × |∫inn

Iout| � |∫inn∗

Iinn∗ |, with

some finite positive number C, where inn∗ (but not inn) intersects hqu, and ‘�’ represents that the LHS

is exponentially suppressed compared to the RHS. The argument establishing a uniform bound is a minor

variation of the one in the next-to-last paragraph of Appendix A of [16].23As well as in the Schur index [38] of SU(N) N = 4 theory [16].

– 22 –

for dimhqu > 0 using the decomposition method applies much more generally to any index of

the form (2.1).

If we are interested only in the singular asymptotics of log I, from (2.57) it follows that,

irrespective of which patches dominate, we have

log I(p, q;v) = −Vout

2∗β2− V out

1∗β

+ dimhqu log( 1

β

)+O(β0), (2.58)

with V out2,1∗ the value of V out

2,1 (x) on hqu. In particular, the singular asymptotics of log I is

entirely determined by the functions V out2,1 (x).

We can summarize our approach and findings as follows.

• We start by decomposing hcl to an outer patch and finitely-many inner patches. See

Definitions 1 and 3.

• On each patch we use uniform small-β estimates (see (2.12) and (2.31)) to simplify the

integrand of the index. This amounts to an efficient representation of the asymptotic

contribution of each patch as in (2.39). (The contribution of the outer patch follows

from (2.39) as explained below (2.41).)

• We find the dominant patches and simplify their contribution by determining hqu. See

Definition 2. Adding up those contributions we get the final asymptotic expression

in (2.57).

Most of the analysis is rather universal. The parts that are model-dependent and should

be carried out on a case by case basis are: i) the specific decomposition of hcl, and ii) the

determination of hqu. We will illustrate these model-dependent aspects of the analysis in the

case study of the SU(N) N = 4 theory index in the next subsection.

Note that more than the specific decomposition of hcl and the resulting fine structure (i.e.

the number and shapes) of the various patches, it is hqu that is important. This is because

according to (2.58) the singular asymptotics of log I can be obtained without any knowledge

of the fine structure of the patches.

What about non-Lagrangian theories?

Our derivation of the asymptotic formula (2.57) relied on the elliptic hypergeometric integral

representation of the index. Such a representation is not available for general non-Lagrangian

theories. But it seems reasonable to expect that the basic structure of the asymptotics would

remain the same for any QFT for which the index can be defined, including non-Lagrangian

theories. More precisely, we suspect that in the limit

β → 0+, with ω1,2 ∈ H, ξa ∈ R fixed, (2.59)

– 23 –

general 4d N = 1 QFTs with a U(1)R symmetry would display asymptotics of the form

I(p, q;v) ' exp

(A

β2+B

β+ βEsusy

)P (

1

β), (2.60)

with P (1/β) a (finite-degree) polynomial in 1/β.

In Lagrangian cases the degree of P is bounded above by the rank of the gauge group,

because dimhqu < dimhcl = rG. It would be interesting if an upper bound can be found for

the degree of P in non-Lagrangian cases as well.

2.5 Example: the N = 4 index

We now proceed to illustrate various model-dependent aspects of the analysis in the specific

case of SU(N) N = 4 theory.

The elliptic hypergeometric integral expression for the index of SU(N) N = 4 theory

reads [43]:

I(p, q; v1,2) =

((p; p)(q; q)

)N−1

N !

3∏a=1

ΓN−1e

((pq)

13 va) ∮ N−1∏

j=1

dzj2πizj

i 6=j∏1≤i,j≤N

∏3a=1 Γe

((pq)

13 va

zizj

)Γe(zizj

) ,

(2.61)

with the unit-circle contour for the zj = e2πixj , while∏Nj=1 zj = 1. In an N = 1 language the

N = 4 theory has an SU(3) flavor symmetry, and v1,2,3 satisfying v1v2v3 = 1 are the fugacities

associated to this flavor symmetry. Note from the formula that each chiral multiplet of the

N = 4 theory is charged (with unit charge) under only one of the va.

2.5.1 Decomposition of the classical moduli space

We refer to hcl as the classical moduli space. In the present case

hcl = (−1

2,1

2]N−1. (2.62)

The starting point of the decomposition scheme of the previous subsection is the singular

set S introduced in (2.15), which for the SU(N) N = 4 theory becomes

S =⋃

1≤i<j≤N{x ∈ hcl|xi − xj ∈ Z}

⋃a=1,2,3

⋃1≤i 6=j≤N

{x ∈ hcl|xi − xj + ξa ∈ Z}. (2.63)

Let us consider the SU(2) case as an example. Then

hcl = (−1

2,1

2], S = {0 , 1

2}

⋃a=1,2,3

{∓ξa2

mod 1 ,1

2∓ ξa

2mod 1}, (2.64)

where by mod 1 we mean the representative in hcl. In particular, in the simple special case

with ξ1,2,3 = 0, we have S = {0 , 12}.

The outer patch S ′ε is the subset of hcl in which for all a and all i 6= j we have minn∈Z(|xi−xj + ξa − n|) ≥ ε, and also minn∈Z(|xi − xj − n|) ≥ ε. In the SU(2) case with ξ1,2,3 = 0 for

example, we have S ′ε = [−12 + ε

2 , −ε2 ] ∪ [ ε2 ,

12 −

ε2 ].

– 24 –

Figure 1. The decomposition of the classical moduli space hcl for SU(2) N = 4 theory with ξ1,2,3 = 0.

In the SU(2) case hcl is parametrized by x1 ∈ [− 12 ,+

12 ] with − 1

2 and + 12 identified. For ε < εc = 1

2 ,

there are two inner patches, and the outer patch consists of two connected components.

The inner patches are the subsets of Sε(= hcl \S ′ε) distinguished from each other by their

differing sets of light modes. In the SU(2) example with ξ1,2,3 = 0, we have two inner patches,

each an open interval of length ε, one around 0 and the other around 12 (note that 1

2 and −12

are identified). See Figure 1.

2.5.2 The quantum moduli space and the asymptotics of the index

We refer to hqu as the quantum moduli space. As discussed above, the patches that intersect

hqu give the dominant contributions to the index in the limit (2.8).

The quantum moduli space coincides with the locus of minima of ReV out2 , or when this

locus is extended, with the subset of that locus where ReV out1 is minimized. For SU(N) N = 4

theory the potentials can be found from (2.18) to be

V out2 (x) =

(2π)3i

6ω1ω2

3∑a=1

[∑i<j

(B3(ξa + xij) +B3(ξa − xij)

)+ (N − 1)B3(ξa)

], (2.65a)

V out1 (x) =

(2π)2i (ω1 + ω2)

4ω1ω2

(− 1

3

3∑a=1

[∑i<j

(B2(ξa + xij) +B2(ξa − xij)

)+ (N − 1)B2(ξa)

]

+ 2∑i<j

B2(xij) +N − 1

6

), (2.65b)

V out0 (x) =

2πi

ω1ω2

((ω1 + ω2

)272

− ω21 + ω2

2

24

) 3∑a=1

(∑i 6=j

B1

(xij + ξa

)+ (N − 1)B1

(ξa)). (2.65c)

The x-dependent piece of ReV out2 , in turn, is of the form

∑i<j Vh(xij), with Vh(x) the

– 25 –

pairwise holonomy potential, which up to a positive overall factor reads24

Vh(x) = sign(Re( −i

(b+ ik1)(b−1 + ik2)

)) 3∑a=1

(B3(ξa + x) +B3(ξa − x)

). (2.66)

We emphasize that although B3 is piecewise cubic, due to the (gauge)3 anomaly cancellation

Vh ends up being piecewise quadratic.

Let us assume k1,2 > 0 for the moment. Since ξ3 is redundant thanks to the balancing

condition v1v2v3 = 1, we can study the pairwise potential Vh(x) as a function of only two

control-parameters ξ1,2 ∈ R. The behavior of Vh(x), and consequently the precise asymptotic

behavior of the index, depend on where on this space of control-parameters we are [18, 21].

As shown in Figure 2, in essentially half of the ξ1 − ξ2 plane the pairwise potential is M-

shaped, while in the mirror half it is W-shaped. There is also a measure-zero bifurcation set

(indicated by the dashed lines in the figure) on which the potential undergoes catastrophic

changes [44]: it either vanishes (on the boundary of the butterfly) or develops plateaux (on

the boundaries of the middle triangles inside the wings). When instead k1,2 < 0, the M and

W wings of Figure 2 switch places.

Finally, the supersymmetric Casimir energy (2.19) for SU(N) N = 4 theory becomes

Esusy =−i(ω1 + ω2)3

54ω1ω2

(N2 − 1

). (2.67)

M wing

We now assume ξ1,2 are chosen such that we land strictly inside the M wing of the ξ1 − ξ2

plane. In this case, the pairwise potential Vh(x) (2.66) has a minimum at x = 0. Then, from

Definition 2 for the quantum moduli space, and from the fact that the x-dependent piece of

ReV out2 is of the form

∑i<j Vh(xij), we conclude that the quantum moduli space hqu of the

SU(N) N = 4 theory on the M wing corresponds to the set of isolated points satisfying xij = 0

for all i, j ∈ {1, · · · , N}. In other words, the holonomies attract and hqu corresponds to all

xj on top of each other, and equal to a given multiple of 1/N due to the SU(N) constraint∑Ni=1 xi ∈ Z.

Importantly, the quantum moduli space intersects the inner patch in0 and its N − 1

images under the ZN center symmetry xj → xj + 1N [18]. We denote these N inner patches

by inn∗ .

The inner patch in0 contributes to the index as

Iin0(p, q; v1,2) ' exp

[− iπστ

(N2 − 1)

(3∏

a=1

({ξa}+

σ + τ

3− 1 + η

2

)+η(σ2 + 3στ + τ2)

12

)]

×(2πβ )N−1

N !

∫in0

N−1∏`=1

dx`√−ω1ω2

e−iπηNστ

∑Nj=1 x

2j

∏1≤i<j≤N

1

Γh(2π(xi−xj)

β )Γh(2π(xj−xi)

β ),

(2.68)

24To compare with [18] note that the complex parameters b, β in that work are related to the real parameters

here as bthere =√

b+ik1b−1+ik2

, βthere = β√

(b+ ik1)(b−1 + ik2).

– 26 –

Figure 2. The qualitative—or catastrophic—profile of the pairwise potential for the holonomies,

Vh(x), drawn over the range x ∈ (−1, 1) of the pair’s separation, for fixed ξ1,2 and fixed k1,2 > 0,

in the two complementary wings of the space of the control-parameters ξ1,2. (Note that ξ1,2 are 1-

periodic: ξ1,2 ∼ ξ1,2 + 1, so the two wings cover a fundamental domain.) The dashed lines indicate

the bifurcation sets across which the qualitative behavior changes. In particular, on the upper-right

wing the potential is M-shaped and the holonomies attract each other, while on the lower-left wing

it is W-shaped and the holonomies repel [18]. (Note that the holonomies are also 1-periodic.) The

qualitative change on a single wing is more fine-grained—less catastrophic—and was not emphasized

in [18]: it only involves appearance or disappearance of plateaux in the potential. The plateaux on

the W wing will be associated to the breakdown of the non-renormalization of log I in Section 4. The

M and W wings switch places if k1,2 are taken to be negative instead.

inside the M wing. This is obtained by specializing the formula (2.39) to the N = 4 theory.

Here we have defined η ∈ {−1,+1} following [23] via

η := 2∑a

{ξa} − 3. (2.69)

Defining the rescaled variables σ` := 2πx`/β, we can rewrite (2.68) as

Iin0(p, q; v1,2) ' exp

[− iπστ

(N2 − 1)

(3∏

a=1

({ξa}+

σ + τ

3− 1 + η

2

)+η(σ2 + 3στ + τ2)

12

)]

× Z3d(ω1, ω2;2πε

β),

(2.70)

– 27 –

where

Z3d(ω1, ω2; Λ) =1

N !

∫|σi−σj |<Λ

N−1∏`=1

dσ`√−ω1ω2

e− iπηNω1ω2

∑Nj=1 σ

2j

∏1≤i<j≤N

1

Γh(σi − σj)Γh(σj − σi),

(2.71)

and with σj subject to∑N

j=1 σj = 0.

We observe that Z3d(ω1, ω2; Λ) coincides with the partition function25 of 3d N = 2 SYM

with gauge group SU(N) and Chern-Simons coupling

kij = −ηNδij , (2.72)

with Coulomb branch cut-off Λ, on a 3-manifold with ω1, ω2 as its moduli of the transversely

holomorphic foliation [46].

Since hqu consists of a set of isolated points, as argued around (2.47) sending ε→∞ on

the RHS of (2.70) introduces exponentially small error:

Z3d(ω1, ω2;2πε

β) ' Z3d(ω1, ω2;∞) (on the M wings). (2.73)

Moreover, since the index is dominated by the patches inn∗ intersecting hqu, we have

I(p, q; v1,2) '∑n∗

Iinn∗ = N Iin0(p, q; v1,2) (on the M wings), (2.74)

with the factor of N accounting for the images of in0 under the ZN center symmetry.

Putting the two findings in (2.73) and (2.74) together and using the asymptotics (2.70),

we can simplify the asymptotic of the index on the M wing as

I(p, q; v1,2) ' N exp

[− iπστ

(N2 − 1)

(3∏

a=1

({ξa}+

σ + τ

3− 1 + η

2

)+η(σ2 + 3στ + τ2)

12

)]× Z3d(ω1, ω2),

(2.75)

where Z3d(ω1, ω2) := Z3d(ω1, ω2;∞).

To make contact with the result of Lezcano et al [23], we use

1

Γh(x;ω1, ω2)Γh(−x;ω1, ω2)= 4 sinh

( πx

−iω1

)sinh

( πx

−iω2

)(2.76)

inside the integrand of Z3d(ω1, ω2), specialize to σ = τ (and hence ω1 = ω2 = ω), define the

re-scaled variable σ′` = σ`−iω , and rotate the contour of σ′` back to the real axis to arrive at

Z3d(ω1, ω2) =1

N !

∫ +∞

−∞

N−1∏`=1

dσ′` eiπηN

∑Nj=1 σ

′2j

∏1≤i<j≤N

4 sinh2(π(σ′i − σ′j)

). (2.77)

25Compare with the expressions in Section 5 of [45], noting that the three-manifold considered there has

ω1 = ib, ω2 = ib−1. To compare with the transversely holomorphic foliation (THF) moduli in [22] note that

the 3d metric there differs by an overall factor from the one in [30] that we are adopting here.

– 28 –

This coincides (up to a convention-dependent sign) with the S3 partition function of the

SU(N)−ηN Chern-Simons theory as encountered in [23].

We emphasize that while the expression (2.77) is simpler looking, as we will see in the

next section it is in fact (2.71) that arises more naturally in the EFT perspective.

The index is fully deconfined on the M wing, both in the sense that the xij = 0 con-

figurations maximally break the ZN center symmetry, and in the sense that the maximal

asymptotic growth is achieved on the M wing [21].

Note that the above derivation of the asymptotics is more rigorous than the one in [23]

because we use the real-analytic approach of Rains [36] and avoid subtle steepest-descent

contour deformation arguments.

Note also that in the preliminary analysis of the leading asymptotics of the 4d N = 4

index on the M wings in [18] some subtleties were not properly treated (see footnote 7 and

the paragraph above Eq. (2.6) in that work). In particular, the estimate (2.12) was used

there, which is not uniformly valid for vector multiplet gamma functions near the dominant

holonomy configurations xij = 0. Furthermore, it was not proven there that phase oscillations

in the integrand do not lead to cancellations (in other words, that completely destructive

interference does not occur). The derivation above overcomes those subtleties because the

estimates we use here are all-order exact and uniformly valid near xij = 0. Our analysis thus

fills the gaps in the derivation of the leading asymptotics of the index in [18].

W wing

Determination of the quantum moduli space on the W wing is difficult for general N [21].

Here we focus on the simple case of SU(2).

As can be seen from Figure 2, on the middle triangle of the W wing, the quantum moduli

space hqu corresponds to x12 = 2x1 = ±1/2, and therefore consists of two isolated points:

x1 = ±1/4. Strictly inside the middle triangle, these points are in the outer patch, and hence

as discussed below (2.25) we have

I(p, q; v1,2) ' 2× eβEsusy e−Vout(x∗) Z

x∗

3d (ω1, ω2), (2.78)

with x∗ = ±1/4. The exponential terms in (2.78) can be written more explicitly as

βEsusy − V out∗ = − πi

2στ

3∏a=1

({2ξa}+

2(σ + τ)

3− 1− η

2

)

+πi

στ

3∏a=1

({ξa}+

σ + τ

3− 1 + η

2

)+

(σ2 + 3στ + τ2)ηπi

4στ.

(2.79)

This is obtained by substituting x12 = ±12 into (2.65) and then using (2.17) and (2.67). We

have also used that

23∑

a=1

{2ξa} − 3 = −η (2.80)

– 29 –

in the middle triangle of the W wing.

Strictly inside the peripheral triangles of the W wing, Figure 2 implies that Vh is min-

imized on an extended domain. More precisely, inside the triangle closest to the origin the

locus of minima of Vh(x12) is

|x12 ±1

2| ≤ 1

2+ ξ1 + ξ2, (2.81)

while inside the left-most triangle the locus is

|x12 ±1

2| ≤ −1

2− ξ1, (2.82)

and inside the lower-most triangle it is

|x12 ±1

2| ≤ −1

2− ξ2. (2.83)

Combining (2.81), (2.82), and (2.83) together, the locus of minima of Vh(x12) can be written

compactly as

|x12 ±1

2| ≤ 1/2− {ξmin} = 1− {1/2 + ξmin}, (2.84)

where ξ3 is determined by the constraint∑3

a=1 ξa ∈ Z modulo an integer, and ξmin is defined

through

{ξmin} := min[{ξa} | a = 1, 2, 3]. (2.85)

The quantum moduli space is the subset of this extended locus where V out1 is minimized.

However, it turns out that V out1 is flat on the locus of minima of Vh on all the three peripheral

triangles. Therefore the quantum moduli space hqu coincides with the loci (2.84). The length

of the quantum moduli space, denoted |hqu|, is then given from (2.84) as

|hqu| = 2(1− {1/2 + ξmin}). (2.86)

We emphasize that hqu is parametrized by the first holonomy x1 (recall that the second

holonomy x2 is determined modulo an integer by the SU(2) constraint x1 + x2 ∈ Z).

Note that the quantum moduli space hqu given in (2.84) intersects the outer patch as well

as inner patches. Therefore we must sum the contributions from all those patches to derive

the asymptotics of the index.

First, the contribution to the index from the intersection of the quantum moduli space

and the outer patch, namely hqu ∩ S ′ε, reads

1

2!

2π

β√−ω1ω2

(|hqu| − 2ε) eβEsusy e−Vout∗ , (2.87)

similarly to (2.28). Note that because V out2,1 are flat on hqu ∩ S ′ε, the coefficients kout

ij , koutjR

are zero and the partition function Z3dn∗(ω1, ω2; 2πε

β ) is particularly simple here; compare with

the discussion around (2.29). The 2ε subtracted above from |hqu| arises because hqu has two

connected components, and by intersecting it with S ′ε we are excising two intervals of length

ε/2 from the ends of each component. The 1/2! is the 1/|W | factor in the present SU(2) case.

– 30 –

Next, the contributions from the four inner patches at the ends of the two components

of hqu read

4×eβEsusy

∫inn

Dx

(√−ω1ω2)rG

(2π

β

)rG×(e−V

inn (x)

∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Ln Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) − e−V out(x)

),

+ 4×eβEsusy

∫inn

Dx

(√−ω1ω2)rG

(2π

β

)rG e−V out(x),

(2.88)

where we have added and subtracted the last line similarly to (2.49). We have also taken

advantage of the symmetries of the problem to take into account the contributions of the four

patches by multiplying the contribution from one of them by four.

Let us now consider the expression inside the large parenthesis in (2.88). According to

(2.37) the first term approaches the second term exponentially fast as we move away from the

end points of the two components of hqu. Therefore we can take out the e−Vout

factor and

call the remaining function inside the parenthesis g(2π(x−x∗)β ), with x∗ the position of the end

point. Since g(2π(x−x∗)β ) approaches zero exponentially fast as we move away from the end

points, we can re-scale x = 2π(x− x∗)/β, and get for the asymptotics of the first integral in

(2.88) the following expression:

eβEsusy e−Vout∗ D1, (2.89)

with

D1 = 4

∫ ∞−∞

dx/2!√−ω1ω2

g(x) e−∆V out(x), (2.90)

where ∆V out(x) := V out(x)− V out∗ .

The second integral in (2.88) has contributions from inside hqu, which cancel the ε-

dependent piece of (2.87). It also has contributions from outside of hqu, which since V out

decays away from hqu can be written as

eβEsusy e−Vout∗ D2, (2.91)

with

D2 = 4

∫ ∞0

dx/2!√−ω1ω2

e−∆V out(x). (2.92)

Putting all the above contributions (2.87), (2.89), and (2.91) together we get

I(p, q; v1,2) ' eβEsusy e−Vout∗ (

2π

β√−ω1ω2

|hqu|2!

+ C0), (2.93)

with C0 := D1 +D2. This is of course a special case of (2.57).

– 31 –

The exponential terms in (2.93) can be written more explicitly as

βEsusy − V out∗ = − πi

2στ

3∏a=1

({2ξa}+

2(σ + τ)

3− 1 + η

2

)

+πi

στ

3∏a=1

({ξa}+

σ + τ

3− 1 + η

2

)− (σ2 + 3στ + τ2)ηπi

12στ.

(2.94)

This is obtained by substituting x12 = ±12 into (2.65) and then using (2.17) and (2.67). We

have also used that

2

3∑a=1

{2ξa} − 3 = η (2.95)

in the peripheral triangles of the W wing.

Here we compare the expression (2.93) for the Cardy-like asymptotics of the N = 4

SU(2) superconformal index in the peripheral triangles of the W -wing with the result from

the Bethe-Ansatz approach [26]. From (3.30) of [26], the Cardy-like asymptotics of N = 4

SU(2) index is given in the peripheral triangles of the W -wing as26

I(q, q; v1,2)

'(i(1− {1/2 + ξmin})

τ− iη

6+

1

2π

)e−

πi2τ2

∏3a=1({2ξa}+

4τ3− 1+η

2 )+ πiτ2

∏3a=1({ξa}+

2τ3− 1+η

2 )− 5ηπi12 ,

(2.96)

where ∆therea = σ+τ

3 + ξherea and we take p = q (σ = τ , hence ω1 = ω2 = ω) and ξa ∈ R for

simplicity. The Cardy-like asymptotics from the Bethe-Ansatz approach (2.96) is consistent

with our result (2.93) under the identification σ = τ = βω2π . First, it is straightforward to

see that the exponential term in (2.93), which is given as (2.94), reduces to the exponential

term in the Bethe-Ansatz result (2.96) under the identification σ = τ . Next, note that the

coefficients of the 1β -order in the polynomial parts of (2.93) and (2.96) match precisely since

the length of the quantum moduli space is given as |hqu| = 2(1 − {1/2 + ξmin}) as we saw

in (2.86). We leave the calculation of the constant C0 in (2.93) and its comparison with the

counterpart − iη6 + 1

2π in (2.96) for future research.

The bifurcation set

The bifurcation set consists of the boundary of the butterfly in Figure 2, as well as the

boundaries of the middle triangles inside the two wings. Actually, the boundary of the middle

triangle on the M wing is completely irrelevant for our purposes, as it is the structure of the

minima of Vh that determines hqu and significantly affects the asymptotics of the index.

On the boundary of the butterfly V out2 vanishes. For simplicity we focus on the origin of

the ξ1 − ξ2 plane, but there is no particular difficulty in extending the analysis to the other

parts of the boundary region.

26In the convention of [26], assuming positive [negative] k1,2 corresponds to η1 = 1 [η1 = −1] in the W -wing,

and the peripheral [middle] triangles correspond to the region with η1 = η2 [η1 = −η2]. Also note ηthere1 = ηhere.

– 32 –

At the origin of the ξ1 − ξ2 plane not only V out2 but also V out

1 vanishes. As a result we

have hqu = hcl: the quantum moduli space coincides with the classical moduli space.

Focusing on the SU(2) case for simplicity again, there are two singular points at x =

0, 1/2, and hence two inner patches as can be seen from Figure 1. These are related to each

other via the center symmetry, so we can consider only in0 and multiply its contribution by

two to account for in1 as well.

The contribution of the outer patch is, as in (2.87), given by

1

2!

2π

β√−ω1ω2

(1− 2ε) eβEsusy e−Vout0 , (2.97)

because in this case |hqu| = |hcl| = 1, while V out1,2 = 0, and V out

0 is independent of x.

The contribution of the inner patches is, similarly to (2.88), given by

2×eβEsusy

∫in0

Dx

(√−ω1ω2)rG

(2π

β

)rG×(e−V

inn (x)

∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Ln Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) − e−V out(x)

),

+ 2×eβEsusy

∫in0

Dx

(√−ω1ω2)rG

(2π

β

)rG e−V out(x).

(2.98)

The first integral can be argued as in (2.89) to be given by

eβEsusy e−Vout0 C0, (2.99)

with

C0 = 2

∫ ∞−∞

dx/2!√−ω1ω2

h(x). (2.100)

Here h(x), with x = 2πx/β, is the exponentially decaying function arising from the second

line of (2.98) near x = 0. The second integral in (2.98) simply kills the ε-dependent part of

(2.97). Putting everything together we hence get

I(p, q; v1,2) ' eβEsusy e−Vout0 (

2π

β√−ω1ω2

1

2!+ C0). (2.101)

Again, this is a special case of (2.57).

3 Effective field theory perspective

The asymptotic formulas in the previous section were derived by direct asymptotic analysis

of the integral expression (2.1), which in turn is obtained via Hamiltonian operator counting

arguments [12, 43]. In this section we present a more physical derivation of the asymptotic

– 33 –

expressions in a path-integral picture via the supersymmetric 3d EFT machinery pioneered

by Di Pietro-Honda-Komargoski [5, 6], and developed to all orders in [22, 30].

Here we refine the treatments in the latter two references by combining their respective

advantages. In particular, we adopt the more general (and more EFT friendly) supersym-

metric background of [30], but follow the more systematic Wilsonian treatment of [22] (which

unlike the approach in [30] can capture partially-deconfined or confined phases as well). We

also add the novel ingredient of decomposition of the BPS moduli space (see Section 3.2),

which we believe completes the conceptual framework.

The 4d background

The starting point of the EFT approach is the Lagrangian formulation of the index I(p, q;v)

as a path-integral partition function Z(p, q;v) on a Hopf surface that is topologically S3×S1

[29, 30, 47]. The metric of the Hopf surface reads [30]

ds24 =

1

1 + b−2k21 cos2 θ + b2k2

2 sin2 θ

[dt2E +

(b2 cos2 θ + b−2 sin2 θ

)dθ2

+ b−2 cos2 θ (dϕ1 + k1dtE)2 + b2 sin2 θ (dϕ2 + k2dtE)2]. (3.1)

Here θ ∈ [0, π/2], the angles ϕ1,2 are 2π-periodic, and the Euclidean time coordinate has the

independent periodicity

tE ∼ tE + β . (3.2)

The complex-structure moduli of the Hopf surface (3.1) are

σ =iβ

2π(b+ ik1), τ =

iβ

2π(b−1 + ik2). (3.3)

This is why the parameters b, β, k1,2 were defined in the previous section via (2.7).

In the approach of [27], to preserve supersymmetry on the Hopf surface one has to turn

on various background supergravity fields, specifically the V and A fields of new minimal

supergravity. We skip their detailed expressions here and refer the interested reader to [30].

On the other hand, the fugacities va correspond to turning on background gauge fields

A(f)a =

2πξaβ

dtE , (3.4)

for the Cartan of the flavor symmetry.

Localization in 4d and the exact partition function

On the 4d background described above, we place our dynamical 4d N = 1 vector and chiral

multiplets. We then employ appropriate supersymmetric Lagrangians constructed as in [27],

and impose periodic boundary conditions along the S1.27 The partition function Z of the

27This corresponds to setting n0 = 0 in [30]. More generally, one can consider twisted identifications of the

form ψ(tE + β) = eπin0(r+F )ψ(tE), with r the U(1)R charge and F the fermion number of the field ψ.

– 34 –

dynamical fields can then be computed exactly via supersymmetric localization of the path-

integral—see [1, 28, 29, 47]. Assuming rχ > 0 as we have in this work, it turns out that on the

localization (or BPS) locus all the dynamical chiral and vector multiplet fields (including the

vector multiplet auxiliary field D) are zero, except the tE component of the vector field [29]:

Ajon BPS locus−−−−−−−−→ 2πxj

βdtE , (3.5)

with xj ∈ (−12 ,

12 ]. Note that using the large gauge transformations (winding the S1) the

Cartan components are restricted to (−πβ ,

πβ ], implying xj ∈ (−1

2 ,12 ]. A Weyl redundancy

remains, which we do not fix at this stage, and instead incorporate into the measure of the

localized path-integral. Thus

hcl = (−1

2,1

2]rG is the moduli space of 4d BPS field configurations,

up to the Weyl redundancy.

For the case of our interest here, with ξa real and hence va on the unit circle, it was

argued in [18] (based on [29, 33]) that the localized partition function Z is given by28

Z(p, q;v) = e−βEsusy I(p, q;v), (3.6)

with I the matrix integral in (2.1), and Esusy as in (2.19). In particular, the integral in (2.1)

is now seen to arise as the integral over the moduli space of BPS field configurations, and the

division by |W | in (2.2) incorporates the Weyl redundancy. The rest of the matrix integral

as well as the SUSY Casimir factor arise from one-loop contributions of the dynamical vector

and chiral multiplets [18, 29, 33, 47].

The goal of the present section

Rather than going the exact 4d localization route of Eq. (3.6), our aim in this section is to

derive the asymptotics of Z via 3d localization in the small-β effective field theory.

To be more concrete, recall that the central result of the previous section was the formula

(2.39) for the Cardy-like asymptotics of Iinn (the contribution of the inner patch inn to the

index); everything else followed from it via elementary manipulations. That asymptotic

formula, according to Eq. (3.6), is equivalent to

Zinn(p, q;v) '∫

inn

D(2πxβ )

(√−ω1ω2)rG

e−Vinn (x)

∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 ) + 2πβ (ρχ · x+ qχ · ξ)Z

)∏α+∈Ln Γh

(2πβ (α+ · x)Z

)Γh(

2πβ (−α+ · x)Z

) .

(3.7)

Reproducing this expression via localization in 3d EFT is the main goal of the present section.

28For general 4d N = 1 indices without flavor fugacities (i.e. with va = 1) the correct exponential factor was

obtained in [33, 37]. For real-valued flavor fugacities a generalization was given in [48]. (That generalization

is presumably valid only in a neighborhood of va = 1, although the details are not properly understood.) In

the present context, with flavor fugacities complex and on the unit circle, it was argued in [18] that the result

of [33, 37] applies.

– 35 –

3.1 The 3d background

Since we are interested in the Cardy-like limit where the circle size β shrinks, we put the

Hopf surface metric (3.1) in a Kaluza-Klein (KK) form

ds24 = ds2

3 + (dtE + c)2. (3.8)

In the three-dimensional picture we then have a manifold M3 with metric

ds23 =

(b2 cos2 θ + b−2 sin2 θ

)dθ2 + b−2 cos2 θ dϕ2

1 + b2 sin2 θ dϕ22

1 + b−2k21 cos2 θ + b2k2

2 sin2 θ

−(b−2k1 cos2 θ dϕ1 + b2k2 sin2 θ dϕ2

1 + b−2k21 cos2 θ + b2k2

2 sin2 θ

)2

,

(3.9)

as well as a graviphoton

c =b−2k1 cos2 θ dϕ1 + b2k2 sin2 θ dϕ2

1 + b−2k21 cos2 θ + b2k2

2 sin2 θ. (3.10)

The KK reduction of the 4d background V and A fields of new minimal supergravity

leads to the 3d background fields v, A(R), and H, which together with the 3d metric comprise

the bosonic components of the 3d new minimal supergravity multiplet. We recite [30]:

H =i+ b−1k1 + bk2√b2 cos2 θ + b−2 sin2 θ

,

v = −i ∗3 dc =2i(k2 cos2 θ dϕ1 + k1 sin2 θ dϕ2

)(1 + b−2k2

1 cos2 θ + b2k22 sin2 θ

)√b2 cos2 θ + b−2 sin2 θ

,

(3.11)

and refer the interested reader to [30] for the explicit expression of A(R) (denoted A there).

The 4d background gauge field A(f) a sits in a 4d N = 1 vector multiplet with auxiliary

field D(f) a = 0. Its KK reduction leads to a 3d background scalar σ(f) a = A(f) atE

as well

as a 3d background gauge field A(f) aµ = −A(f) a

tEcµ, which sit together with an auxiliary field

D(f) a = −A(f) atE

H in a 3d N = 2 vector multiplet. See [6, 22].

We emphasize that all 4d N = 1 multiplets—the background multiplets whose dimen-

sional reduction was just described as well as the dynamical multiplets whose KK expansion

will be described below—yield 3d N = 2 multiplets on M3.

3.2 Light fields and cut-offs

After discussing the dimensional reduction of the background fields, we now consider the KK

expansion of the dynamical fields.

The KK expansion leads to infinite towers of fields. This applies to all the 4d fields in the

dynamical 4d N = 1 vector and chiral multiplets. If all the fields in these towers are kept,

we are essentially back in the 4d realm; indeed the “4d localization computation” is actually

– 36 –

done in this way [29]: by KK expanding the 4d fields, localizing the path-integrals of the 3d

fields in the KK towers, and then summing over the towers.

Instead, we can integrate out the “heavy” 3d fields in the KK towers, and keep only the

“light” fields. This leads to a Wilsonian approximation of the full partition function, which

as we will see reproduces the asymptotics up to exponentially small corrections.

What makes the Wilsonian procedure particularly rich in the supersymmetric context

is that often supersymmetric partition functions are integrals over a moduli space, and the

separation of dynamical fields into “light” and “heavy” can not be done uniformly over all of

that moduli space. In the present setting, we will see that different parts of the BPS moduli

space support different sets of light and heavy fields.

One of the main ideas of the present paper is that the EFT explanation of the asymp-

totics of the 4d index should proceed by decomposing the moduli space hcl of 4d BPS field

configurations into various patches, and assigning to each patch its own 3d EFT according to

the set of light fields the patch supports.

A clear sense of which 3d fields are light and which are heavy, is provided by the asymp-

totic hierarchy of scales emerging in the β → 0 limit.

Therefore two notions of “cut-off” arise in our analysis. The first one is the Wilsonian

cut-off which sets the boundary between light and heavy 3d fields. We denote this cut-off by

ΛM , and express it in terms of β explicitly as

ΛM =2πε

β. (3.12)

Here ε is some β-independent, positive number. A specific choice of ε amounts to a precise

scheme for separation of mass scales.

The second notion is that of the field-space cut-offs which set the boundaries between

neighboring patches of the moduli space. A specific choice of these cut-offs amounts to a

precise scheme for decomposition of the moduli space.

The particular scheme that we adopt here for decomposition of hcl is based on the set

of 3d fields that are light on each patch. Here by “light” we mean having mass less than

the cut-off ΛM . In this way, we hence link our field-space decomposition scheme to our

mass-separation scheme, so that our decomposition scheme is also fixed when we pick a

specific value for ε. We emphasize though that this decision to link so closely the field-space

decomposition and mass-separation schemes is motivated by the technical requirements of

the present problem, and may not be as useful in other contexts.

Here we discuss the separation of 3d fields into light and heavy explicitly only for the

fermions. The 3d N = 2 supersymmetry guarantees that each light fermion is accompanied

by its scalar and/or vector super-partners, and similarly for the heavy fields.

The 3d fermion arising from the nth KK mode of the weight ρχ of a 4d chiral multiplet

χ would have real-mass (in the sign convention of [5])

m[n]ρχ =

2πn

β− 2π ρχ · x

β− 2π qχ · ξ

β, (3.13)

– 37 –

with the notation as in the previous section. This is light if

|m[n]ρχ | < ΛM ⇐⇒ |n− ρχ · x− qχ · ξ| < ε, (3.14)

and heavy otherwise. Below we will denote the real-mass of the light 3d fermion arising from

the weight ρχ by mρχ . Note that

mρχ = −2π

β(ρχ · x+ qχ · ξ)Z. (3.15)

On the other hand, denoting the roots of the gauge group by α, the 3d fermion arising

from the nth KK mode of the α component of the vector multiplet would have real-mass

m[n]α =

2πn

β− 2π α · x

β. (3.16)

This is light if

|m[n]α | < ΛM ⇐⇒ |n− α · x| < ε, (3.17)

and heavy otherwise. Below we will denote the real-mass of the light 3d fermion arising from

the root α by mα. Note that

mα = −2π

β(αχ · x)Z. (3.18)

The outer patch

Recall that in our convention rG of the roots are zero, and correspond to the Cartan of

the gauge group. We can see from (3.17) that the n = 0 mode of the Cartan fermions are

light—indeed massless—for all x.

If there are zero weights ρχ in the problem, we call them “light” only if Qχ(ξ) := qχ ·ξ ∈ Z(because otherwise we can take ε small enough so that (3.14) is not satisfied for any n). In

other words, a weight ρχ is called light if it yields a light 3d fermion as in (3.14). We can

see from (3.14) that the n = Qχ(ξ) mode of the fermions in these light zero weights are also

light—massless indeed—for all x.

The outer patch, from the EFT perspective, is the subset of the BPS moduli space

hcl where there are no light 3d fermions besides the massless fermions of the previous two

paragraphs. The massless fermions are of course accompanied by their 3d N = 2 super-

partners. In particular, the massless fermions arising from the 4d N = 1 vector multiplet sit

in massless 3d N = 2 gauge multiplets whose bosonic field content is the gauge field Aµ, the

Coulomb branch scalar σ, and the auxiliary field D.

In the outer patch we thus have a 3d N = 2 gauge theory with U(1)rG gauge group

arising from the n = 0 mode of the Cartan of the 4d vector multiplet, together with decoupled

(ρχ = 0) massless 3d N = 2 chiral multiplets arising from the n = Qχ(ξ) mode of the light

zero weights of the 4d chiral multiplets.

– 38 –

Anticipating our 3d localization computation below, we note that (as emphasized in the

present context in [6]) on the BPS locus of the 3d N = 2 U(1)rG gauge theory we have

σj = AjtEon BPS locus−−−−−−−−→ 2πxj

β,

Ajµ = Ajµ −AjtEcµ

on BPS locus−−−−−−−−→ −2πxjβ

cµ,

Dj = Dj −AjtEHon BPS locus−−−−−−−−→ −2πxj

βH.

(3.19)

Inner patches

Inner patches are the subsets of hcl where additional light 3d fermions arise. They can be

distinguished from their neighboring patches by their differing set of light fermions. Note

that in a patch inn the 4d fermion associated to a root α (or a weight ρχ) yields a light 3d

fermion only if α ∈ Ln (or ρχ ∈ Ln). This explains our terminology in the previous section

for Ln as the set of “light” roots and weights. Of course 4d fermions associated to light roots

and weights yield infinite KK towers of heavy 3d fields as well.

The extra light fermions of the inner patches also sit inside 3d N = 2 multiplets. In

particular, if they arise from KK expansion of the 4d gauginos, they would be accompanied

by light 3d vector fields that may combine with the massless Cartan gauge fields to enhance

the gauge group of the low energy EFT from U(1)rG to a non-abelian subgroup of G.

For example, in the inner patch in0, not just the Cartan but all components of the 4d

gauge field yield light 3d vector fields through their n = 0 mode. This means that on in0

we have a 3d N = 2 gauge theory with gauge group G. If there is a center symmetry (as in

SU(N) N = 4 theory) images of in0 under the center symmetry will also have gauge group G.

These patches may or may not have additional light chiral multiplets as well.

There may also be patches where the gauge group is a proper non-abelian subgroup of G.

Those are the patches that dominate the index in partially-deconfined phases [21, 49, 50].

We emphasize that there could be inner patches where the EFT has the same U(1)rG

gauge group as the outer patch, but with additional light chiral multiplets arising from the

4d chiral multiplets. In other words, the gauge symmetry of the 3d EFT is not necessarily

enhanced on inner patches.

On those inner patches where the gauge group is enhanced, we have to deal with non-

abelian 3d vector multiplets. But anticipating our 3d localization computation below, we

know that we need not worry about the non-Cartan components, because the BPS locus even

in the non-abelian case is (3.19).

Since in0 is the most special inner patch, it is worth noting that in terms of the 3d

Coulomb branch scalar we can characterize in0 as (set n = 0 in (3.17) and use (3.19))

in0 : |α · σ| < ΛM . (3.20)

This is a sharp instance of the close relation between the mass-separation scheme and the

field-space decomposition scheme.

– 39 –

3.3 The Wilsonian effective action

The tree-level action of the light 3d fields, denoted SLtree, comes from the UV: in the mode

expansion of the 4d action we keep only the terms associated to the light fields. This is

straightforward if one has access to the 4d Lagrangian on the Hopf surface, which can be

found in [29] (based on [27, 51, 52]).

The tree-level action is corrected at the one-loop level by the contribution arising from

integrating out the heavy fields. The calculation needed for directly finding the complete one-

loop correction is difficult (indeed daunting) because of the various kinds of fields present in

our problem. The breakthrough realization of Di Pietro and Komargodski [5] (refined further

in [6, 22, 30]) was that one can recover the total result indirectly as follows. First, we compute

certain one-loop generated Chern-Simons couplings that are easily under control and receive

contributions only from the heavy fermions. Next, we supersymmetrize the resulting Chern-

Simons actions. This accounts indirectly for the one-loop contribution of the heavy bosonic

fields (including the KK modes of the dynamical gauge field and its ghosts) as well. Although

a complete argument that the resulting supersymmetrized Chern-Simons actions exhaust the

one-loop corrections is not currently available, as in [22, 30] we assume the exhaustion and

find results that are in agreement with the previous section, save for a technical gap explained

below (see also Problem 1 in Section 4).

The one-loop induced SUSY CS action is of the form (cf. [6, 22])

δS1-loop =∑

f

(Sf

g-g + 2Sfg-R + Sf

R-R + Sfgrav + Sf

v

), (3.21)

where the sum is over all the heavy fermions in the theory. Here Sfg-g stands for the supersym-

metrized gauge-gauge CS action induced by integrating out a heavy fermion, together with

the supersymmetrized gauge-KK and KK-KK actions that always accompany it. We also

incorporate the background gauge fields A(f)a into the same term following [6], as elaborated

on in Appendix A. The term 2Sfg-R stands for the supersymmetrized gauge-R CS action, to-

gether with the supersymmetrized KK-R term that always accompanies it, while the factor

of 2 accounts also for the supersymmetrized R-gauge and R-KK terms. The next two actions

on the RHS of (3.21) are the supersymmetrized versions of RR and gravitational CS actions.

The last action on the RHS of (3.21) is new—it was not discussed in [6, 22, 30]. It

stands for a yet-to-be-discovered supersymmetrized CS action in 3d new minimal supergravity,

containing (mixed) CS terms involving v (such as v∧dv) but no RR or gravitational CS term.

We have introduced it to resolve a difficulty (see the discussion below (A.34)) that was not

addressed in [6, 22, 30]. It may also provide a resolution to a mismatch puzzle raised in [30]

(see Eq. (A.42) in the appendix, and also Problem 1 in Section 4).

Assuming that the heavy fermion arises from the nth KK mode of a 4d chiral multiplet

– 40 –

χ, we have on the BPS locus (3.19) that (cf. [22])

Sfg-g

on BPS locus−−−−−−−−→ − i

8πsgn(m

[n]ρχ) (2π(n− ρχ · x− qχ · ξ)

β

)2

AM3 ,

Sfg-R

on BPS locus−−−−−−−−→ − i

8πsgn(m

[n])ρχ) (2π(n− ρχ · x− qχ · ξ)

β

)· (rχ − 1) LM3 ,

SfR-R = − i

8πsgn(m

[n]ρχ) (

(rχ − 1)2 − 1

6

)RM3 ,

Sfgrav = − i

192πsgn(m

[n]ρχ)GM3 .

(3.22)

Here we have written the supersymmetrized CS actions as the one-loop CS coupling (see

Appendix A.1) times the supersymmetrization of an∫A ∧ dA type term. That is except for

the RR CS term whose coefficient is shifted by −1/6 to cancel the extra RR term arising from

the supersymmetrized gravitational CS term. The precise form of AM3 , LM3 , RM3 , and GM3

can be found in Appendix A.2. We actually need to evaluate them on the 3d background of

Section 3.1, and the resulting expressions can be found in Appendix A.2 as well.

We now perform the sum over the heavy 3d fermions. For the 3d fermions arising from

a 4d chiral multiplet χ this involves a sum over the weights ρχ as well as a sum over the KK

numbers n. For the weights ρχ ∈ Hn we sum over all n ∈ Z, and use∑n∈Z

sgn(n+ x)(n+ x)j−1 = −2

jBj(x) , (3.23)

to evaluate the KK sums. For the weights ρχ ∈ Ln we sum over all n ∈ Z except for the

integer that yields a light 3d multiplet. Then use

′∑n∈Z

sgn(n+ x)(n+ x)j−1 = −2

jKj(x) , (3.24)

which follows from (3.23) and (2.32), to simplify the result. The prime in the above sum

means that the integer n for which |n+ x| < ε is excluded from the summation.

The inclusion of the heavy 3d fermions arising from the nth KK mode of a 4d vector

multiplet leads to expressions similar to (3.22):

Sfg-g

on BPS locus−−−−−−−−→ − i

8πsgn(m[n]α

) (2π(n− α · x)

β

)2

AM3 ,

Sfg-R

on BPS locus−−−−−−−−→ − i

8πsgn(m[n]α

) (2π(n− α · x)

β

)LM3 ,

SfR-R = − i

8πsgn(m[n]α

) (1− 1

6

)RM3 ,

Sfgrav = − i

192πsgn(m[n])α

)GM3 .

(3.25)

Then the sum over 3d heavy fermions from the KK modes of a 4d vector multiplet can also

be done using the formulas (3.23) and (3.24).

– 41 –

Incorporating (3.22) and (3.25) and then performing the sums over the heavy fermions,

we get ∑f

Sfg-g

on BPS locus−−−−−−−−→ V inn2 (x)

β2,

∑f

2Sfg-R

on BPS locus−−−−−−−−→ V inn1 (x)

β,

∑f

(Sf

R-R + Sfgrav + Sf

v

) != V inn

0 (x),

(3.26)

with V inn2,1,0(x) as in (2.41). Note that in absence of a precise expression for Sf

v, we are

conjecturing here that its incorporation would lead to the last relation above. (An alternative

form of this conjecture is (A.42)). From (2.40) and (3.21), we then have

δS1-loopon BPS locus−−−−−−−−→ V inn(x). (3.27)

Some arguments in [30] suggest that there are no higher-loop corrections to the Wilsonian

effective action. Although a complete non-renormalization argument directly in the 3d picture

is not currently available, we assume it in the following and will find a perfect match with

the results of the previous section. The effective action that we use for supersymmetric

localization of the light 3d fields is hence

Seff = SLtree + δS1-loop. (3.28)

In summary, the UV action is the tree-level 4d action, which can be KK expanded and

expressed in terms of infinitely-many 3d fields. We integrate out the heavy 3d fields and obtain

an IR effective action for the light 3d fields. Note that we do not integrate out the high-energy

modes of the light 3d fields (with “energy” as measured via some time direction inside M3,

and “high” meaning above the cut-off ΛM ); as the following localization calculation shows it

is useful here not to further amputate the high-energy tail of the light 3d fields.

3.4 Localization in the 3d EFT and the asymptotics of the partition function

Supersymmetric localization of the 3d EFT on M3 gives

ZEFTinn

(p, q;v) =

∫e−S

Ltree−δS1-loop

localization−−−−−−−→∫

2πβ

inn

Dσ

(√−ω1ω2)rG

∏χ

∏ρχ∈Ln Γh

(rχ(ω1+ω2

2 )−mρχ)∏

α+∈Ln Γh(−mα+

)Γh(mα+

) e−Vinn (x).

(3.29)

The e−Vinn (x) factor arises from δS1-loop evaluated on the BPS locus. The rest is the familiar

expression (see e.g. [46] or Section 5 of [45]) arising from localization of SLtree, assuming (based

on expectations from holomorphy and dimensional reduction) that ω1, ω2 are the moduli of

the transversely holomorphic foliation [46] of M3.

– 42 –

The above result (3.29) matches our asymptotic expression (2.39) upon using the explicit

expressions for the light real-masses in (3.15) and (3.18), as well as translating between σ

and x via (3.19).

4 Discussion

An interplay between asymptotic analysis and EFT description of exact results was explored

in this work.

When the exact result is a localized supersymmetric partition function and the limit of

interest introduces a scale hierarchy in the problem, an important aspect of the said interplay

is that asymptotic estimates of the integrand lose uniform validity precisely on those parts

of the BPS moduli space where additional light fields emerge that were not incorporated (or

were treated as heavy) in the estimate. This calls for a decomposition of the moduli space

into various patches according to the set of light fields they support, and using appropriate

estimates with uniform validity over each patch.

In the previous sections we fleshed out the details of this procedure in the specific context

of the Cardy-like limit of the 4d superconformal index. We gleaned in particular the following

two lessons that have the promise of finding wider applicability in asymptotic analysis of

supersymmetric partition functions:

• that competition between patches of the BPS moduli space can be a more powerful per-

spective than competition between saddles, especially in presence of extended/degenerate

saddles (this is the perspective that in Section 2 enabled us to establish the polynomial

factor in the general formula (2.57));

• that not only the quantum moduli space as a whole, but also its intersections with

the strata of the singular set of the classical moduli space (as a topologically stratified

space) can be of significance for asymptotics of the partition function (see the discussion

around (2.53)).

Below we highlight another lesson, as well as a few puzzles arising from our study.

4.1 Non-renormalization: Wilsonian effective action versus log I

We saw in the previous section that compatibility of the EFT arguments with the asymptotic

analysis of Section 2 leads to the conclusion that the Wilsonian effective action (3.28) does

not receive higher-loop corrections; in other words, it is exact to all orders in 1/ΛM . (A direct

EFT demonstration of this result is desirable; see below.)

When dimhqu = 0, or more generally when the polynomial factor in (2.57) has only

one non-zero term, the non-renormalization of Seff implies that the small-β expansion of

logZ(p, q;v) terminates at O(β0)—and consequently the small-β expansion of log I(p, q;v)

terminates at O(β) with the linear term being βEsusy.

– 43 –

But when dimhqu > 0 and the polynomial factor in (2.57) contains more than one term,

it is easy to see that the small-β expansion of logZ(p, q;v)—or log I(p, q;v)—does not ter-

minate. This is analogous to the standard statement in supersymmetric QFT that while

the (F -term part of the) Wilsonian effective action is not renormalized beyond one loop, the

(F -term part of the) 1PI action can receive corrections at higher orders.

A non-trivial aspect of our findings in this context, unexpected from standard super-

symmetric QFT literature, is that an extended quantum moduli space—or presence of mass-

less degrees of freedom—is not sufficient for breakdown of the non-renormalization of log I(i.e. termination of its small-β expansion). The latter requires that the polynomial factor

in (2.57) has at least two nonzero terms; i.e. that Cj = Coutj (ε) +

∑n∗C

inn∗j (ε) 6= 0 for at

least two different values of the non-negative integer j, with Coutj (ε) and C

inn∗j (ε) introduced

in (2.30) and (2.54) respectively. Roughly speaking, the index should receive dominant con-

tributions from regions of the moduli space that localize in the β → 0 limit to subsets with

different dimensions.

This sheds new light on why inclusion of decoupled massless abelian vector multiplets—

which give rise to an extended quantum moduli space—does not spoil the termination of

the small-β expansion of log I [33]. That is because the index of decoupled abelian vector

multiplets is a product of Pochhammer symbols, so they multiply the polynomial factor of

(2.57) only by a monomial (see (2.11)). Therefore if that polynomial had only one nonzero

term before their inclusion, it will have only one nonzero term after their inclusion as well,

and the small-β expansion of log I still terminates.

4.2 Open problems

We have left a few technical issues in our analysis unaddressed. For example, in comparing

the asymptotics of the N = 4 SU(2) index on the peripheral triangles of the W wing as

follows from the Bethe-Ansatz approach and as follows from our integral based approach, we

have not demonstrated that the coefficient C0 in (2.93) equals − iη6 + 1

2π (see below (2.96)).

Another technical issue is that we have not demonstrated that theM3 metric in (3.9) admits

a transversely holomorphic foliation (THF) with moduli ω1,2 as in (2.9).

More importantly, our asymptotic analysis has led to formulation of sharp and concep-

tually interesting puzzles in Wilsonian EFT. We highlight three of them below, as examples

of EFT projects that are motivated and informed by asymptotic analysis of exact results.

Problem 1) Demonstrate that the SUSY CS terms exhaust the one-loop corrections to the

Wilsonian effective action of Section 3, at least on the BPS locus.

In other words, prove that no other SUSY action is generated at one loop besides the SUSY

CS terms, or that if such extra actions are generated they vanish on the BPS locus so that

the localization result (3.29) is unaltered.

Note that the result (3.29) relied on the conjectural equality on the third line of (3.26).

So part of the above problem is to identify the SUSY CS action Sv in (3.21), and to show

– 44 –

that once evaluated on the supersymmetric background of Section 3 it leads to the equality

in (3.26) (which otherwise would not be valid for general ω1,2 ∈ H according to the discussion

at the end of Appendix A.2).

Problem 2) Prove a non-renormalization theorem in the EFT of Section 3, prohibiting

higher-loop corrections to Seff, at least on the BPS locus.

In other words, prove that the effective action (3.28) is exact to all orders in 1/ΛM , at least

on the BPS locus. Such a theorem would be a curved super-space counterpart of the standard

F -term non-renormalization theorem on flat space [53].

Problem 3) Why is zeta-function regularization appropriate for the KK sums featuring in

the 3d effective action?

The matching between asymptotic analysis and EFT also relied on our use of the zeta-

function regularization for the infinite KK sums arising in the 3d EFT—see (3.23). It would

be desirable to have a satisfactory argument directly in the EFT to justify this use. (See

Appendix D of [54] for a proof of appropriateness of the zeta-function regularization in a

related context. There, non-commutativity of the small-β limit and plethystic exponentiation

necessitates regularization of the plethystic exponential of a small-β expansion. Here, non-

commutativity of the small-β limit and KK expansion necessitates regularization of the KK

sum over small-β contributions.)

Acknowledgments

This paper is in many ways a follow-up to our earlier works [23] and [22]. We would like

to acknowledge the essential contribution of our collaborators in those projects, A. Gonza-

lez Lezcano, J. Liu, S. Murthy, and L. Pando Zayas, to the ideas that are developed further

in this work. We are also grateful to D. Cassani for sharing with us the computations lead-

ing to (A.41), and to A. Bourget, A. Cabo Bizet, J. Grimminger, A. Hanany, and Z. Zhong

for related discussions. AA began this project at KCL where he was supported by the

ERC Consolidator Grant N. 681908, “Quantum black holes: A macroscopic window into the

microstructure of gravity”, and is now supported by the NSF grant PHY-1915093 and the

Simons Foundation (Simons Collaboration on the Non-Perturbative Bootstrap, grants 397411

and 488647). JH is supported in part by an Odysseus grant G0F9516N from the FWO, by the

KU Leuven C1 grant ZKD1118 C16/16/005, and by the Research Programme of the Research

Foundation grant G.0926.17N from the FWO.

A Supersymmetric Chern-Simons actions

In this appendix, we consider 3d effective Chern-Simons (CS) terms generated by integrating

out heavy fermions arising from the Kaluza-Klein (KK) expansion of a 4d N = 1 chiral

multiplet. We also discuss supersymmetrization of the CS terms.

– 45 –

A.1 Chern-Simons actions from integrating out heavy fermions

We start from the relevant part of the Euclidean 4d action of an N = 1 chiral multiplet,

namely (M,N ∈ {x1, x2, x3, tE} are 4d coordinate indices)

SUV ⊇ SχUV =

∫d4x√g

[iχσMDMχ+

1

2VM χσMχ

](A.1)

where the covariant derivative is given as

DM = ∇M − iρχ ·AM − iqχ ·A(f)M − i(rχ − 1)A

(R)M , (A.2)

and χ stands for the fermion field in the chiral multiplet labeled by the same symbol. Here

we have mainly followed the convention of [29]. To be specific, ρχ is the weight of the rep-

resentation of a chiral multiplet with respect to the gauge symmetry and rχ is the R-charge

of the chiral multiplet (the R-charge of the fermion in the multiplet is therefore rχ − 1). qχ

is the vector of charges under various flavor symmetries. AM stands for the Cartan of the

dynamical gauge field, and A(f)M is the background gauge field associated with flavor symme-

tries; the latter was not included in [29], but it is necessary for capturing flavor fugacities.

A(R)M and VM are the R-symmetry background gauge field and the auxiliary 1-form in the

gravity multiplet of the 4d new minimal supergravity [55, 56]. The sigma matrices and the

connection ∇M are given in our convention as (A,B ∈ {1, 2, 3, 4} are 4d frame indices)

σM = (~σ,−iI2), σM = (−~σ,−iI2), ∇Mχ = (∂M −1

4ωABM σ[AσB])χ. (A.3)

Combining (A.2) and (A.3), we can rewrite (A.1) explicitly as

SχUV =

∫d4x√g

[iχσM (∂M −

1

4ωABM σ[AσB] − iYM )χ

], (A.4)

where we have introduced a 4d 1-form YM as

YM ≡ ρχ ·AM + qχ ·A(f)M + (rχ − 1)A

(R)M +

1

2VM . (A.5)

Next we explore the KK compactification of the 4d action (A.4) on S1 of the 4d manifold

that is topologically M3 × S1, and whose metric is of the form (µ, ν ∈ {x1, x2, x3} are 3d

coordinate indices)

ds24 = gMNdx

MdxN = hµνdxµdxν + (dtE + cµdx

µ)2, (A.6)

where tE ∼ tE + β. Using (a, b ∈ {1, 2, 3} are 3d frame indices)

(e−1)MA =

((e−1

(3d))µa 0

−cµ(e−1(3d))

µa 1

), ωab4 = −1

2fab, ω4b

a = −1

2fab, (A.7)

– 46 –

in terms of the field strength of a KK photon, fab ≡ ∂acb − ∂bca, the 4d action (A.4) can be

rewritten as

SχUV =

∫dtE d3x

√h

[−χiσa

((e−1

(3d))µa∂µ − cµ(e−1

(3d))µa∂tE +

1

4ωbca σ[bσc] +

i

4fabσb

− i(e−1(3d))

µaYµ + icµ(e−1

(3d))µaYtE

)χ

+χ(∂tE +1

8fabσ[aσb] − iYtE )χ

].

(A.8)

Substituting the KK expansion of the 4d fermion

χ =∑n∈Z

1√βe2πnitE/βχ(n), χ =

∑n∈Z

1√βe−2πnitE/βχ(n), (A.9)

and integrating over the x4 coordinate, the 4d action (A.8) reads

SχUV =∑n∈Z

∫d3x√h

[χ(n)

(− iσa(e−1

(3d))µa∂µ −

i

4ωbca σ

aσbσc +1

8fabσaσb + im

[n]ρχ

− σa(e−1(3d))

µa(Y(0)

µ +2πn

βcµ))χ(n)

]+ · · · ,

(A.10)

with the dots arising from the non-zero KK modes of the dynamical gauge field. Above we

have defined a shifted fermion mass m[n]ρχ and a combined 3d gauge field Y(0)

µ as

m[n]ρχ =

2πn

β− Y (0)

tE, Y(0)

µ = Y (0)µ − Y (0)

tEcµ, (A.11)

with the superscripts (0) emphasizing that only the zeroth KK mode of the dynamical gauge

field is included. Note that the superscript “[n]” of the mass m[n]ρχ in (A.11) does not mean

that it is the nth mode of a KK expansion: it just represents that it is a function of n.

We would like to see how CS terms show up in the 3d effective action. For that purpose,

first we write down the partition function of an N = 1 chiral multiplet in terms of the effective

action Seff obtained by integrating out heavy fermions as

Z =

∫[dχdχdY · · · ] exp[−SUV[χ, χ,Y, · · · ]]

=

∫[dY · · · ] exp[−Seff[Y, · · · ]],

(A.12)

where ‘· · · ’ represent other fields in the chiral multiplet. As mentioned above, we first consider

the contribution from the Sχ part given in (A.10) to the effective action Seff, namely Sχeff[Y] ⊆Seff[Y, · · · ]. It is given as

exp[−Sχeff[Y]

]=

∫[dχdχ] exp

[−SχUV[χ, χ,Y]

]. (A.13)

– 47 –

The contribution from other fields to the effective action Seff will then be obtained by su-

persymmetrizing Sχeff. Note that, according to (A.13), Sχeff[Y] can be computed by evaluating

connected Feynman diagrams as

Sχeff[Y] = −⟨exp[−Sχint

]⟩connected, (A.14)

where the interaction term Sχint is read off from (A.10) as

Sχint =∑n∈Z

∫d3x√h

[χ(n)

(− i

4ωbca σ

aσbσc +1

8fabσaσb − σa(e−1

(3d))µa(Y(0)

µ +2πn

βcµ)

)χ(n)

].

(A.15)

Evaluating the effective action (A.14) at one-loop order, we will derive a CS term for the

combined 3d gauge field Y(0)µ . The following calculation involves only elementary QFT but

we provide details for completeness.

To compute the effective action (A.14) at one-loop order explicitly, we consider the sim-

plest case with flat hµν = δµν .29 Then we use the Pauli matrices as 3d Euclidean gamma

matrices, namely γµ ∈ {σ1, σ2, σ3}, which satisfy the trace formulas

tr(γµγν) = 2δµν ,

tr(γµγνγρ) = 2iεµνρ,

tr(γµγργνγσ) = 2(δνσδµρ − δρσδµν + δµσδνρ).

(A.16)

Now the one-loop order effective action (A.14) reads

Sχeff[Y]∣∣1-loop

= − 1

2!

′∑m,n∈Z

∫d3x

∫d3y (Y(0)µ(x) +

2πm

βcµ(x))(Y(0) ν(y) +

2πn

βcν(y))

×⟨χ(m)(x)γµχ

(m)(x)χ(n)(y)γνχ(n)(y)

⟩,

(A.17)

with the prime over the summation symbol excluding the light fermions. Using the Wick’s

theorem, as well as the Fourier transforms of the gauge field Y(0) and the KK photon c, and

also the fermion propagator read off from (A.10) as

⟨χ(m)(x)χ(n)(y)

⟩= δmn

∫d3k

(2π)3

1

γρkρ + im[n]ρχ

eik·(x−y), (A.18)

29A more methodical approach would be to work on curved background and use the heat kernel.

– 48 –

one can rewrite the one-loop order (A.17) as

Sχeff[Y]∣∣1-loop

=1

2!

′∑n∈Z

∫d3x

∫d3y

∫d3p

(2π)3(Y(0)µ(p) +

2πn

βcµ(p)) eip·x

∫d3q

(2π)3(Y(0) ν(q) +

2πn

βcν(q)) eiq·y

×∫

d3k1

(2π)3

∫d3k2

(2π)3eik1·(x−y)+ik2·(y−x) tr

(γµ

1

γρkρ1 + im

[n]ρχ

γν1

γρkρ2 + im

[n]ρχ

).

(A.19)

The overall minus sign comes from having a fermion loop. In (A.19), evaluating the x, y-

integrals gives two Dirac-delta functions. Using one of them, one can evaluate the k2-integral

too and thereby simplify (A.19) as

Sχeff[Y]∣∣1-loop

=1

2!

′∑n∈Z

∫d3p

(2π)3

∫d3q

(2π)3(Y(0)µ(p) +

2πn

βcµ(p))(Y(0) ν(q) +

2πn

βcν(q))

× (2π)3δ3(p+ q)Π[n]ρχµν(p), (A.20a)

Π[n]ρχµν(p) ≡

∫d3k

(2π)3tr

(γµ

1

γρkρ + im[n]ρχ

γν1

γρ(p+ k)ρ + im[n]ρχ

). (A.20b)

Here we compute Π[n]ρχµν(p), omitting the superscript “[n]” and the subscript “ρχ” for

notational convenience. Using the identity∫ 1

0dx

1

((1− x)a+ xb)2=

1

ab, (A.21)

and then changing the integration variable k → l = k + xp, one can simplify (A.20b) as

Πµν(p) =

∫ 1

0dx

∫d3l

(2π)3

Nµν

(l2 + x(1− x)p2 +m2)2,

Nµν = 4lµlν − 2gµν l2 − x(1− x)

(4pµpν − 2gµνp

2)

+ 2mpρεµνρ − 2m2gµν .

(A.22)

Here we have also used the trace formulas (A.16). Using lµlν → 13gµν l

2 upon the l-integration

and the identity∫dnl

(l2)α

(l2 +M2)β=

2πn2

Γ(n2 )(M2)

n2

+α−β Γ(α+ n2 )Γ(β − α− n

2 )

2Γ(β), (A.23)

one can evaluate the l-integral in (A.22) that gives

Πµν(p) =

∫ 1

0dx

[− 2

3(2π)3gµν

2π32

Γ(32)

(x(1− x)p2 +m2)12

Γ(52)Γ(−1

2)

2Γ(2)

+−x(1− x)(4pµpν − 2gµνp

2) + 2mpρεµνρ − 2m2gµν(2π)3

× 2π32

Γ(32)

(x(1− x)p2 +m2)−12

Γ(32)Γ(1

2)

2Γ(2)

].

(A.24)

– 49 –

In the m → ∞ limit, the O(|m|) terms from the first and the second lines of (A.24) cancel

each other. Consequently, we obtain

limm→∞

Πµν(p) =m

|m|1

4πpρεµνρ. (A.25)

Substituting (A.25) back into (A.20a) then gives

limβ→0+

Sχeff[Y]∣∣1-loop

=1

2!

′∑n∈Z

∫d3p

(2π)3

∫d3q

(2π)3(Y(0)µ(p) +

2πn

βcµ(p))(A(0) ν(q) +

2πn

βcν(q))

× (2π)3δ3(p+ q)sgn(m

[n]ρχ)

4πpρεµνρ.

(A.26)

Note that the large mass limit m[n]ρχ →∞ for the heavy fermions is justified by the Cardy-like

limit β → 0+. Finally, using the inverse Fourier transforms of Y(0) and c, and then integrating

by parts we obtain

limβ→0+

Sχeff[Y]∣∣1-loop

= − i

8π

′∑n∈Z

sgn(m

[n]ρχ) ∫

d3x εµνρ(Y(0)µ +

2πn

βcµ)∂ν(Y(0)

ρ +2πn

βcρ). (A.27)

The resulting CS term matches the one in the literature precisely. See [5, 6] for example.

Adding a gravitational CS term (see Appendix A of [57] for example, and refer to foot-

note 32 below for comments on the overall sign)

− i

192π

′∑n∈Z

sgn(m

[n]ρχ) ∫

d3x√gεµνρ Tr

(ωµ∂νωρ − 2

3ωµωνωρ)

(A.28)

to (A.27), and then summing over all weights ρχ for the chiral multiplet χ, we obtain the

total effective CS terms. The result can be written as

Sχ1-loop CS = − i

8π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ) ∫

d3x εµνρ(Y(0)µ +

2πn

βcµ)∂ν(Y(0)

ρ +2πn

βcρ)

− i

192π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ) ∫

d3x√gεµνρ Tr

(ωµ∂νωρ − 2

3ωµωνωρ).

(A.29)

A.2 Supersymmetrization of Chern-Simons terms

One can attempt supersymmetrizing the effective CS terms in (A.29) explicitly as (see [22])

Sχ1-loop = − i

8π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)

(sg-g + 2(rχ − 1)sg-R)

− i

8π

((rχ − 1)2 − 1

6

)∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)sR-R −

i

192π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)sgrav,

(A.30)

– 50 –

in terms of the following 3d supersymmetric actions30

sg-g =

∫M3

[(ρχ · A+ qχ · A(f) +

2πn

βc)∧ d(ρχ · A+ qχ · A(f) +

2πn

βc)

−2i(ρχ · D + qχ · D(f) +

2πn

βH)(2πn

β− ρχ · σ − qχ · σ(f)

)(∗31)

], (A.31a)

sg-R =

∫M3

[(ρχ · A+ qχ · A(f) +

2πn

βc)∧ dA(R) + iH

(ρχ · D + qχ · D(f) +

2πn

βH)(∗31)

− i4

(2πn

β− ρχ · σ − qχ · σ(f)

)(R(3) + 2vµvµ + 2H2

)(∗31)

], (A.31b)

sR-R =

∫M3

[A(R) ∧ dA(R) + i

H

2

(R(3) + 2vµvµ + 2H2

)(∗31)

], (A.31c)

sgrav =

∫M3

[Tr

(ω ∧ dω − 2

3ω ∧ ω ∧ ω

)+ 4(A(R) − v

)∧ d(A(R) − v

)]. (A.31d)

The first two actions (A.31a) and (A.31b) are from Appendix A of [6] with the inclusion of

flavor symmetries31

ρχ · A → ρχ · A+ qχ · A(f), ρχ · σ → ρχ · σ+ qχ · σ(f), ρχ · D → ρχ · D+ qχ · D(f), (A.32)

and the last two actions (A.31c) and (A.31d) are from Appendix A of [57]. 32

To obtain the supersymmetric effective action (A.30) from (A.29) in terms of (A.31), we

have relied on relations between 4d superfields and 3d ones, namely (see Appendix A of [30]

for example)

σ = AtE , Aµ = Aµ −AtEcµ, D = D −AtEH,

σ(f) = A(f)tE, A(f)

µ = A(f)µ −A

(f)tEcµ, D(f) = D(f) −A(f)

tEH,

H = A(R)tE

= VtE , A(R)µ = A(R)

µ −A(R)tEcµ,

1

2vµ = Vµ − VtEcµ,

(A.33)

with vµ = −iεµνρ∂νcρ. In particular, note that the combined one-form Y(0) defined by (A.5)

and (A.11) can be rewritten using (A.33) as

Y(0) = ρχ · A+ qχ · A(f) + (rχ − 1)A(R) +1

4v. (A.34)

It is this expression (A.34) that we use in (A.29) to make contact with the supersymmetric

actions (A.31).

We would like to emphasize that the combination of supersymmetric actions in (A.30)

provides effective CS terms of the dynamical gauge field A, the background gauge field for

30Our A(R) corresponds to A(R)there − v/2 in [6, 22].

31Note that D in Appendix A of [6] should be replaced with D.32In the effective action (A.30), the signs of the CS terms are different from those in [57]. This is due to the

different sign conventions used for fermion real-masses: compare our Eq. (A.11) with (A.1) and (A.11) of [57].

– 51 –

flavor symmetry A(f), the background gauge field for R-symmetry A(R), and the KK photon

c precisely, but does not yield correct (mixed) CS terms for the KK photon field strength v.

In particular, from (A.29) and (A.34) we get a coefficient − i8π

∑ρχ∑′

n∈Z sgn(m

[n]ρχ)× (1/4)2

for∫v ∧ dv, whereas the only

∫v ∧ dv term in (A.30) arises from sgrav and has coefficient

− i192π

∑ρχ∑′

n∈Z sgn(m

[n]ρχ)× 4; the two clearly do not match. The supersymmetric effective

action (A.30) is therefore valid only if v vanishes on the 3d manifold of interest, M3. This

mismatch puzzle was not addressed in [22, 30]. To fix it for the general case with non-

vanishing v, we have introduced the extra (unspecified) SUSY CS action Sv in (3.21). We

leave the precise determination of Sv for future research.

On the BPS locus of the dynamical vector multiplet, and with the specific expression for

the background vector multiplet for flavor symmetry as in

AtE =2πx

β, Aµ = 0, D = 0,

A(f)tE

=2πξ

β, A(f)

µ = 0, D(f) = 0,

(A.35)

the supersymmetric effective action (A.30), with Sv included, reads

Sχ1-loop = − i

8π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)((2π(n− ρχ · x− qχ · ξ)

β

)2

AM3

+2(rχ − 1)

(2π(n− ρχ · x− qχ · ξ)

β

)LM3

)− i

8π

((rχ − 1)2 − 1

6

)∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)RM3 −

i

192π

∑ρχ

′∑n∈Z

sgn(m

[n]ρχ)GM3 + Sv.

(A.36)

Definitions of AM3 , LM3 , RM3 , GM3 are as follows (these four supersymmetric CS actions

involving background fields were studied in a series of papers by Closset and collaborators—

see in particular [57, 58])

AM3 =

∫M3

d3x√g(i vµ cµ − 2 iH

),

LM3 =

∫M3

d3x√g(i vµA(R) − i 1

2vµ vµ + i

1

2H2 − i 1

4R(3)

),

RM3 =

∫M3

d3x√g(εµνρA(R)

µ ∂νA(R)ρ + i

H

2

(R(3) + 2vµv

µ + 2H2))

,

GM3 =

∫M3

d3x√g(εµνρ Tr

(ωµ ∂ν ωρ − 2

3ωµ ων ωρ)

+ 4 εµνρ(A(R)µ − vµ

)∂ν(A(R)ρ − vρ

)).

(A.37)

The first two were discussed in [6]. To compare with [6] note that

AM3 = −π2AthereM3

LM3 = −iπ2LthereM3

. (A.38)

– 52 –

All four have been discussed in [22, 30]. To compare with [30] note that

AM3 = − iβ2

πIthere

1 , LM3 = −iβIthere2 , RM3 = −4πiIthere

3 , GM3 = −192πiIthere4 .

(A.39)

Another notational difference with [30] is that ωthere1,2 = −iβω1,2. We have chosen to work with

the SUSY actions in (A.37) (rather than I1,2,3,4) because they have clear physical significance

as the supersymmetrized KK-KK, R-KK, R-R, and gravitational CS terms, respectively. We

have also chosen to keep the symbols ω1,2 for the THF moduli of M3, to be consistent with

standard references such as [45].

To compare with [22] note that

AM3 = π2AthereM3

, LM3 = −π2LthereM3

, RM3 = π2RthereM3

, GM3 = π2GthereM3

. (A.40)

On the 3d background of Section 3.1, the actions in (A.37) evaluate to [30]:

AM3 = − 4π2

ω1ω2,

LM3 = 2π2 ω1 + ω2

ω1ω2,

RM3 = −π2 (ω1 + ω2)2

ω1ω2.

(A.41)

Moreover, we conjecture

GM3 + · · · != 4π2 (ω1 − ω2)2

ω1ω2, (A.42)

with the · · · standing for the modification that GM3 would receive if the contribution from

Sv is incorporated into it. This conjecture is based on holomorphy, as (A.42) is found to

be valid for a limited range of ω1,2 in [30]. The work [30] reported however that GM3 alone

on the RHS of (A.42) can not reproduce its LHS for general ω1,2 ∈ H, and raised this as a

mismatch puzzle. We are suggesting that incorporating Sv may resolve that mismatch. This

would imply that with the inclusion of Sv the third line in (3.26) is satisfied.

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